what happens when you raise an exponential expression to a power

Learning Objectives

• Product and Quotient Rules
  • Use the product rule to multiply exponential expressions
  • Use the caliber rule to divide exponential expressions
• The Ability Rule for Exponents
  • Utilize the power dominion to simplify expressions involving products, quotients, and exponents
• Negative and Zero Exponents
  • Define and use the zero exponent rule
  • Define and use the negative exponent rule
• Simplify Expressions Using the Exponent Rules
  • Simplify expressions using a combination of the exponent rules
  • Simplify compound exponential expressions with negative exponents

Repeated Epitome

Beefcake of exponential terms

We utilize exponential notation to write repeated multiplication. For example [latex]10\cdot10\cdot10[/latex] can be written more succinctly as [latex]10^{3}[/latex]. The x in [latex]10^{three}[/latex]is called the base. The iii in [latex]10^{three}[/latex]is called the exponent. The expression [latex]10^{3}[/latex] is chosen the exponential expression. Knowing the names for the parts of an exponential expression or term will aid y'all acquire how to perform mathematical operations on them.

[latex]\text{base}\rightarrow10^{three\leftarrow\text{exponent}}[/latex]

[latex]10^{three}[/latex] is read as "10 to the third power" or "x cubed." It means [latex]10\cdot10\cdot10[/latex], or 1,000.

[latex]8^{2}[/latex] is read equally "8 to the second power" or "eight squared." It means [latex]8\cdot8[/latex], or 64.

[latex]5^{4}[/latex] is read as "5 to the fourth ability." It means [latex]5\cdot5\cdot5\cdot5[/latex], or 625.

[latex]b^{5}[/latex] is read as "b to the fifth power." It means [latex]{b}\cdot{b}\cdot{b}\cdot{b}\cdot{b}[/latex]. Its value will depend on the value of b.

The exponent applies only to the number that it is next to. Therefore, in the expression [latex]xy^{4}[/latex], only the y is affected by the 4. [latex]xy^{4}[/latex] means [latex]{x}\cdot{y}\cdot{y}\cdot{y}\cdot{y}[/latex]. The x in this term is a coefficient of y.

If the exponential expression is negative, such as [latex]−3^{iv}[/latex], it means [latex]–\left(3\cdot3\cdot3\cdot3\right)[/latex] or [latex]−81[/latex].

If [latex]−3[/latex] is to be the base, information technology must be written as [latex]\left(−3\right)^{4}[/latex], which means [latex]−3\cdot−3\cdot−iii\cdot−3[/latex], or 81.

Also, [latex]\left(−x\right)^{iv}=\left(−x\correct)\cdot\left(−x\right)\cdot\left(−ten\right)\cdot\left(−x\right)=ten^{4}[/latex], while [latex]−10^{four}=–\left(x\cdot x\cdot x\cdot x\right)[/latex].

You can see that there is quite a difference, and then you accept to be very careful! The following examples prove how to identify the base and the exponent, besides as how to identify the expanded and exponential format of writing repeated multiplication.

Example

Identify the exponent and the base in the following terms, then simplify:

1. [latex]7^{two}[/latex]
2. [latex]{\left(\frac{ane}{2}\right)}^{iii}[/latex]
3. [latex]2x^{iii}[/latex]
4. [latex]\left(-5\right)^{2}[/latex]

In the following video you lot are provided more examples of applying exponents to diverse bases.

Evaluate expressions

Evaluating expressions containing exponents is the same every bit evaluating the linear expressions from before in the course. You lot substitute the value of the variable into the expression and simplify.

Y'all can use the order of operations to evaluate the expressions containing exponents. First, evaluate anything in Parentheses or grouping symbols. Next, await for Exponents, followed by Multiplication and Sectionalisation (reading from left to right), and lastly, Improver and Subtraction (again, reading from left to right).

And so, when you evaluate the expression [latex]5x^{three}[/latex] if [latex]x=4[/latex], kickoff substitute the value 4 for the variable 10. Then evaluate, using society of operations.

Example

Evaluate [latex]5x^{three}[/latex]if [latex]ten=4[/latex].

In the case below, notice the how adding parentheses can alter the event when you are simplifying terms with exponents.

Example

Evaluate [latex]\left(5x\right)^{3}[/latex] if [latex]x=4[/latex].

The addition of parentheses made quite a difference! Parentheses allow you to employ an exponent to variables or numbers that are multiplied, divided, added, or subtracted to each other.

Case

Evaluate [latex]x^{3}[/latex] if [latex]10=−four[/latex].

Caution! Whether to include a negative sign as role of a base of operations or not oft leads to defoliation. To analyze whether a negative sign is practical before or afterwards the exponent, hither is an example.

What is the divergence in the way you would evaluate these two terms?

1. [latex]-{3}^{two}[/latex]
2. [latex]{\left(-iii\right)}^{2}[/latex]

To evaluate 1), y'all would apply the exponent to the three commencement, and so apply the negative sign terminal, like this:

[latex]\begin{array}{c}-\left({3}^{2}\right)\\=-\left(9\right) = -nine\end{array}[/latex]

To evaluate 2), y'all would apply the exponent to the 3 and the negative sign:

[latex]\begin{array}{c}{\left(-three\correct)}^{ii}\\=\left(-3\correct)\cdot\left(-3\right)\\={ nine}\end{assortment}[/latex]

The central to remembering this is to follow the order of operations. The start expression does non include parentheses so you lot would apply the exponent to the integer iii first, and so apply the negative sign. The second expression includes parentheses, so hopefully yous will remember that the negative sign also gets squared.

In the next sections, you volition larn how to simplify expressions that comprise exponents. Come back to this page if you forget how to employ the order of operations to a term with exponents, or forget which is the base of operations and which is the exponent!

In the following video you are provided with examples of evaluating exponential expressions for a given number.

Employ the product dominion to multiply exponential expressions

Exponential annotation was adult to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential note when multiplying or dividing. Let's look at rules that will allow yous to do this.

For example, the note [latex]five^{4}[/latex] tin be expanded and written as [latex]5\cdot5\cdot5\cdot5[/latex], or 625. And don't forget, the exponent only applies to the number immediately to its left, unless there are parentheses.

What happens if you multiply two numbers in exponential class with the same base? Consider the expression [latex]{ii}^{3}{ii}^{iv}[/latex]. Expanding each exponent, this can exist rewritten equally [latex]\left(2\cdot2\cdot2\correct)\left(2\cdot2\cdot2\cdot2\right)[/latex] or [latex]2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2[/latex]. In exponential form, you would write the product as [latex]2^{seven}[/latex]. Notice that seven is the sum of the original two exponents, three and 4.

What about [latex]{x}^{ii}{x}^{six}[/latex]? This can be written as [latex]\left(x\cdot{x}\right)\left(ten\cdot{x}\cdot{ten}\cdot{ten}\cdot{x}\cdot{ten}\right)=ten\cdot{x}\cdot{x}\cdot{x}\cdot{ten}\cdot{x}\cdot{x}\cdot{x}[/latex] or [latex]ten^{eight}[/latex]. And, one time again, eight is the sum of the original two exponents. This concept tin be generalized in the post-obit way:

The Product Rule for Exponents

For any number 10 and any integers a and b, [latex]\left(x^{a}\right)\left(x^{b}\right) = ten^{a+b}[/latex].

To multiply exponential terms with the same base, add together the exponents.

Circumspection! When you are reading mathematical rules, information technology is of import to pay attending to the conditions on the dominion.  For example, when using the product rule, y'all may only apply it when the terms being multiplied have the same base of operations and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this instance it says "For whatever number x, and any integers a and b."

Instance

Simplify.

[latex](a^{iii})(a^{7})[/latex]

When multiplying more than complicated terms, multiply the coefficients and then multiply the variables.

Instance

Simplify.

[latex]5a^{4}\cdot7a^{half dozen}[/latex]

Caution! Do not effort to apply this dominion to sums.

Remember about the expression [latex]\left(2+3\correct)^{2}[/latex]

Does [latex]\left(2+3\correct)^{2}[/latex] equal [latex]two^{2}+3^{2}[/latex]?

No, it does not because of the order of operations!

[latex]\left(2+3\right)^{ii}=5^{two}=25[/latex]

and

[latex]2^{2}+3^{two}=4+9=13[/latex]

Therefore, you tin only use this rule when the numbers within the parentheses are beingness multiplied (or divided, every bit we will meet adjacent).

Use the quotient rule to divide exponential expressions

Allow's look at dividing terms containing exponential expressions. What happens if you lot divide ii numbers in exponential form with the same base? Consider the following expression.

[latex] \displaystyle \frac{{{4}^{5}}}{{{four}^{2}}}[/latex]

You can rewrite the expression equally: [latex] \displaystyle \frac{iv\cdot 4\cdot four\cdot iv\cdot 4}{iv\cdot 4}[/latex]. Then you can cancel the mutual factors of iv in the numerator and denominator: [latex] \displaystyle [/latex]

Finally, this expression tin exist rewritten as [latex]4^{three}[/latex] using exponential note. Notice that the exponent, 3, is the deviation between the two exponents in the original expression, 5 and 2.

So, [latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{two}}}=iv^{5-two}=four^{3}[/latex].

Be conscientious that you subtract the exponent in the denominator from the exponent in the numerator.

And so, to separate two exponential terms with the same base, subtract the exponents.

The Quotient (Partition) Rule for Exponents

For any non-zero number x and any integers a and b: [latex] \displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[/latex]

Example

Evaluate. [latex] \displaystyle \frac{{{4}^{9}}}{{{4}^{4}}}[/latex]

When dividing terms that also contain coefficients, divide the coefficients and then split up variable powers with the same base by subtracting the exponents.

Case

Simplify. [latex] \displaystyle \frac{12{{x}^{iv}}}{2x}[/latex]

In the following video we show another example of how to utilise the quotient rule to carve up exponential expressions

Raise powers to powers

Another word for exponent is power.  You take probable seen or heard an example such as [latex]3^v[/latex] tin can be described as 3 raised to the 5th ability. In this section nosotros will farther aggrandize our capabilities with exponents. Nosotros will larn what to do when a term with a ability is raised to another power, and what to do when ii numbers or variables are multiplied and both are raised to an exponent.  We will also learn what to do when numbers or variables that are divided are raised to a power.  We will begin by raising powers to powers.

Let'southward simplify [latex]\left(5^{2}\correct)^{4}[/latex]. In this case, the base is [latex]5^2[/latex]and the exponent is four, so you multiply [latex]5^{2}[/latex]iv times: [latex]\left(five^{ii}\right)^{4}=v^{2}\cdot5^{2}\cdot5^{2}\cdot5^{ii}=5^{8}[/latex](using the Production Rule—add the exponents).

[latex]\left(5^{2}\right)^{4}[/latex]is a power of a power. It is the fourth power of five to the second ability. And we saw above that the answer is [latex]5^{8}[/latex]. Notice that the new exponent is the same equally the production of the original exponents: [latex]2\cdot4=8[/latex].

So, [latex]\left(5^{2}\right)^{iv}=five^{ii\cdot4}=v^{8}[/latex] (which equals 390,625, if you do the multiplication).

Likewise, [latex]\left(x^{iv}\correct)^{3}=x^{4\cdot3}=x^{12}[/latex]

This leads to another rule for exponents—the Ability Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For instance, [latex]\left(2^{iii}\correct)^{5}=2^{fifteen}[/latex].

The Power Dominion for Exponents

For any positive number x and integers a and b: [latex]\left(ten^{a}\right)^{b}=x^{a\cdot{b}}[/latex].

Have a moment to contrast how this is different from the product rule for exponents found on the previous page.

Example

Simplify [latex]6\left(c^{4}\right)^{2}[/latex].

Raise a product to a power

Simplify this expression.

[latex]\left(2a\right)^{4}=\left(2a\correct)\left(2a\correct)\left(2a\correct)\left(2a\right)=\left(2\cdot2\cdot2\cdot2\right)\left(a\cdot{a}\cdot{a}\cdot{a}\cdot{a}\right)=\left(ii^{4}\right)\left(a^{iv}\right)=16a^{4}[/latex]

Notice that the exponent is applied to each gene of 2a. So, nosotros can eliminate the center steps.

[latex]\begin{array}{l}\left(2a\right)^{4} = \left(2^{iv}\right)\left(a^{4}\right)\text{, applying the }4\text{ to each factor, }ii\text{ and }a\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,=16a^{four}\end{array}[/latex]

The product of ii or more numbers raised to a power is equal to the product of each number raised to the same ability.

A Product Raised to a Power

For any nonzero numbers a and b and any integer x, [latex]\left(ab\right)^{10}=a^{ten}\cdot{b^{10}}[/latex].

How is this rule different from the ability raised to a power rule? How is it different from the product rule for exponents on the previous page?

Example

Simplify. [latex]\left(2yz\right)^{6}[/latex]

If the variable has an exponent with it, utilize the Ability Rule: multiply the exponents.

Example

Simplify. [latex]\left(−7a^{four}b\right)^{ii}[/latex]

Enhance a quotient to a power

Now let's await at what happens if you heighten a quotient to a power. Call back that quotient means divide. Suppose yous have [latex] \displaystyle \frac{3}{iv}[/latex] and raise information technology to the 3^rd power.

[latex] \displaystyle {{\left( \frac{3}{four} \right)}^{3}}=\left( \frac{iii}{4} \right)\left( \frac{3}{4} \right)\left( \frac{3}{iv} \right)=\frac{iii\cdot 3\cdot three}{4\cdot 4\cdot 4}=\frac{{{3}^{3}}}{{{iv}^{3}}}[/latex]

You tin see that raising the quotient to the power of 3 tin also exist written as the numerator (3) to the power of 3, and the denominator (4) to the power of 3.

Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to ability.

[latex] \displaystyle {{\left( \frac{a}{b} \right)}^{four}}=\left( \frac{a}{b} \correct)\left( \frac{a}{b} \correct)\left( \frac{a}{b} \right)\left( \frac{a}{b} \correct)=\frac{a\cdot a\cdot a\cdot a}{b\cdot b\cdot b\cdot b}=\frac{{{a}^{4}}}{{{b}^{4}}}[/latex]

When a quotient is raised to a power, you can apply the ability to the numerator and denominator individually, as shown beneath.

[latex] \displaystyle {{\left( \frac{a}{b} \right)}^{4}}=\frac{{{a}^{iv}}}{{{b}^{4}}}[/latex]

A Quotient Raised to a Power

For any number a, whatever non-nil number b, and any integer ten, [latex] \displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{ten}}[/latex].

Case

Simplify. [latex] \displaystyle {{\left( \frac{ii{x}^{2}y}{ten} \right)}^{iii}}[/latex]

In the following video you will exist shown examples of simplifying quotients that are raised to a power.

Define and apply the zero exponent dominion

When we defined the quotient dominion, we only worked with expressions like the following: [latex]\frac{{{4}^{9}}}{{{4}^{4}}}[/latex], where the exponent in the numerator (upwardly) was greater than the one in the denominator (down), so the terminal exponent after simplifying was always a positive number, and greater than goose egg. In this section, nosotros volition explore what happens when nosotros apply the quotient dominion for exponents and become a negative or nothing exponent.

What if the exponent is aught?

To run across how this is defined, let the states begin with an example. We volition use the idea that dividing any number past itself gives a result of one.

[latex]\frac{t^{8}}{t^{viii}}=\frac{\cancel{t^{8}}}{\cancel{t^{viii}}}=1[/latex]

If we were to simplify the original expression using the caliber dominion, we would accept

[latex]\frac{{t}^{8}}{{t}^{8}}={t}^{eight - 8}={t}^{0}[/latex]

If nosotros equate the two answers, the result is [latex]{t}^{0}=1[/latex]. This is true for any nonzero real number, or whatsoever variable representing a real number.

[latex]{a}^{0}=1[/latex]

The sole exception is the expression [latex]{0}^{0}[/latex]. This appears later in more than avant-garde courses, only for now, we will consider the value to be undefined, or DNE (Does Non Exist).

Exponents of 0 or 1

Any number or variable raised to a power of one is the number itself.

[latex]northward^{1}=n[/latex]

Whatever non-zero number or variable raised to a power of 0 is equal to 1

[latex]n^{0}=ane[/latex]

The quantity [latex]0^{0}[/latex] is undefined.

As done previously, to evaluate expressions containing exponents of 0 or 1, substitute the value of the variable into the expression and simplify.

Instance

Evaluate [latex]2x^{0}[/latex] if [latex]x=9[/latex]

Example

Simplify [latex]\frac{{c}^{three}}{{c}^{3}}[/latex].

In the post-obit video at that place is an example of evaluating an expression with an exponent of zippo, as well as simplifying when you get a result of a zip exponent.

Define and employ the negative exponent dominion

We proposed some other question at the beginning of this department.  Given a quotient like [latex] \displaystyle \frac{{{2}^{1000}}}{{{2}^{n}}}[/latex] what happens when due north is larger than chiliad? We volition demand to use the negative dominion of exponents to simplify the expression so that it is easier to sympathise.

Let's await at an example to clarify this idea. Given the expression:

[latex]\frac{{h}^{3}}{{h}^{5}}[/latex]

Expand the numerator and denominator, all the terms in the numerator volition cancel to i, leaving 2 hsouthward multiplied in the denominator, and a numerator of 1.

[latex]\begin{array}{l} \frac{{h}^{three}}{{h}^{5}}\,\,\,=\,\,\,\frac{h\cdot{h}\cdot{h}}{h\cdot{h}\cdot{h}\cdot{h}\cdot{h}} \\ \,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}}{\cancel{h}\cdot \cancel{h}\cdot \cancel{h}\cdot {h}\cdot {h}}\\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{1}{h\cdot{h}}\\\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\frac{i}{{h}^{2}} \end{array}[/latex]

We could take also practical the quotient dominion from the last section, to obtain the following upshot:

[latex]\begin{array}{r}\frac{h^{3}}{h^{5}}\,\,\,=\,\,\,h^{3-5}\\\\=\,\,\,h^{-2}\,\,\end{assortment}[/latex]

Putting the answers together, we accept [latex]{h}^{-2}=\frac{1}{{h}^{2}}[/latex]. This is true when h, or whatever variable, is a real number and is not zero.

The Negative Rule of Exponents

For whatsoever nonzero real number [latex]a[/latex] and natural number [latex]n[/latex], the negative rule of exponents states that

[latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]

Allow'southward looks at some examples of how this rule applies under different circumstances.

Instance

Evaluate the expression [latex]{4}^{-three}[/latex].

Instance

Write [latex]\frac{{\left({t}^{three}\right)}}{{\left({t}^{8}\right)}}[/latex] with positive exponents.

Example

Simplify [latex]{\left(\frac{i}{3}\right)}^{-2}[/latex].

Instance

Simplify.[latex]\frac{1}{four^{-two}}[/latex] Write your reply using positive exponents.

In the follwoing video you will see examples of simplifying expressions with negative exponents.

Simplify expressions using a combination of exponent rules

Once the rules of exponents are understood, you lot can begin simplifying more than complicated expressions. At that place are many applications and formulas that make use of exponents, and sometimes expressions tin can get pretty cluttered. Simplifying an expression before evaluating tin oftentimes make the computation easier, as you volition run into in the following example which makes use of the quotient rule to simplify before substituting 4 for x.

Case

Evaluate [latex] \displaystyle \frac{24{{x}^{8}}}{2{{10}^{5}}}[/latex] when [latex]x=4[/latex].

Instance

Evaluate [latex] \displaystyle \frac{24{{10}^{8}}{{y}^{2}}}{{{(ii{{x}^{iii}}y)}^{2}}}[/latex] when [latex]x=4[/latex] and [latex]y=-2[/latex].

Detect that you could take worked this trouble by substituting 4 for 10 and 2 for y in the original expression. You would still get the reply of 96, but the computation would be much more circuitous. Notice that you lot didn't even need to use the value of y to evaluate the in a higher place expression.

In the following video you are shown examples of evaluating an exponential expression for given numbers.

Usually, it is easier to simplify the expression before substituting any values for your variables, simply you will go the same answer either manner. In the next examples, yous will see how to simplify expressions using dissimilar combinations of the rules for exponents.

Example

Simplify. [latex]a^{2}\left(a^{v}\correct)^{3}[/latex]

The following examples require the utilise of all the exponent rules nosotros have learned and so far. Remember that the product, power, and quotient rules apply when your terms have the aforementioned base.

Instance

Simplify. [latex] \displaystyle \frac{{{a}^{ii}}{{({{a}^{5}})}^{3}}}{eight{{a}^{8}}}[/latex]

Simplify Expressions With Negative Exponents

Now we will add together the last layer to our exponent simplifying skills and practice simplifying compound expressions that have negative exponents in them. It is standard convention to write exponents equally positive because it is easier for the user to empathize the value associated with positive exponents, rather than negative exponents.

Use the post-obit summary of negative exponents to help you simplify expressions with negative exponents.

Rules for Negative Exponents

Witha, b, m, and n not equal to zero, and mandn as integers, the following rules apply:

[latex]a^{-1000}=\frac{ane}{a^{m}}[/latex]

[latex]\frac{1}{a^{-m}}=a^{m}[/latex]

[latex]\frac{a^{-n}}{b^{-chiliad}}=\frac{b^yard}{a^n}[/latex]

When yous are simplifying expressions that take many layers of exponents, it is ofttimes hard to know where to start. It is mutual to start in one of two ways:

• Rewrite negative exponents as positive exponents
• Apply the product rule to eliminate any "outer" layer exponents such as in the following term: [latex]\left(5y^3\right)^2[/latex]

We will explore this idea with the following case:

Simplify. [latex] \displaystyle {{\left( 4{{x}^{iii}} \right)}^{5}}\cdot \,\,{{\left( 2{{x}^{ii}} \correct)}^{-four}}[/latex]

Write your reply with positive exponents. The table below shows how to simplify the aforementioned expression in two different ways, rewriting negative exponents equally positive first, and applying the product rule for exponents first. You volition encounter that there is a column for each method that describes the exponent rule or other steps taken to simplify the expression.

Rewrite with positive Exponents Kickoff	Clarification of Steps Taken	Utilize the Product Dominion for Exponents Start	Clarification of Steps Taken
[latex] \frac{\left(4x^{3}\right)^{5}}{\left(2x^{two}\right)^{iv}}[/latex]	move the term [latex]{{\left( ii{{10}^{2}} \right)}^{-iv}}[/latex] to the denominator with a positive exponent	[latex] \left(4^5x^{15}\right)\left(two^{-four}10^{-8}\right)[/latex]	Employ the exponent of 5 to each term in expression on the left, and the exponent of -four to each term in the expression on the correct.
[latex]\frac{\left(4^5x^{15}\right)}{\left(2^4x^{8}\right)}[/latex]	Use the product rule to employ the outer exponents to the terms inside each set of parentheses.	[latex]\left(4^v\right)\left(2^{-4}\correct)\left(x^{15}\cdot{x^{-eight}}\correct)[/latex]	Regroup the numerical terms and the variables to make combining like terms easier
[latex]\left(\frac{four^5}{2^4}\right)\left(\frac{x^{15}}{10^{eight}}\correct)[/latex]	Regroup the numerical terms and the variables to make combining like terms easier	[latex]\left(4^5\right)\left(2^{-4}\right)\left(x^{15-viii}\right)[/latex]	Utilise the rule for multiplying terms with exponents to simplify the ten terms
[latex]\left(\frac{iv^5}{2^4}\right)\left(x^{xv-8}\correct)[/latex]	Use the quotient rule to simplify the x terms	[latex]\left(\frac{4^five}{2^4}\right)\left(x^{7}\right)[/latex]	Rewrite all the negative exponents with positive exponents
[latex]\left(\frac{1,024}{16}\right)\left(x^{7}\right)[/latex]	Expand the numerical terms	[latex]\left(\frac{1,024}{16}\right)\left(ten^{7}\right)[/latex]	Aggrandize the numerical terms
[latex]64x^{7}[/latex]	Carve up the numerical terms	[latex]64x^{7}[/latex]	Dissever the numerical terms

If yous compare the ii columns that describe the steps that were taken to simplify the expression, y'all will encounter that they are all about the aforementioned, except the order is changed slightly. Neither way is better or more than correct than the other, it truly is a thing of preference.

Example

Simplify [latex]\frac{\left(t^{3}\right)^2}{\left(t^2\right)^{-8}}[/latex]

Write your answer with positive exponents.

Example

Simplify [latex]\frac{\left(5x\right)^{-2}y}{x^3y^{-i}}[/latex]

Write your answer with positive exponents.

In the adjacent section, you lot will larn how to write very large and very pocket-size numbers using exponents. This practice is widely used in science and engineering.

Summary

• Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.
• The product dominion for exponents: For any number x and any integers a and b, [latex]\left(x^{a}\correct)\left(x^{b}\right) = x^{a+b}[/latex].
• The quotient rule for exponents: For any non-zero number x and any integers a and b: [latex] \displaystyle \frac{{{ten}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[/latex]
• The ability dominion for exponents:
  1. For any nonzero numbers a and b and any integer x, [latex]\left(ab\right)^{x}=a^{10}\cdot{b^{x}}[/latex].
  2. For any number a, whatever not-zero number b, and any integer x, [latex] \displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex]

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Source: https://courses.lumenlearning.com/suny-beginalgebra/chapter/read-terms-and-expressions-with-exponents/

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