The rules of When a number is to express in the kind ab, b is the exponent. The exponent shows how many times the basic is used as a factor. Power and exponent typical the very same thing.

You are watching: Raising a product to a power


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are very useful as soon as simplifying and assessing expressions. When multiplying, dividing, or elevating a strength to a power, making use of the rules because that exponents helps to make the process an ext efficient. Now let’s will certainly look at rules for taking a product or a quotient come a power.


Once the rules of exponents space understood, friend can begin solving more facility expressions an ext easily. Recall that as soon as you take it a strength to a power, you main point the exponents, (xa)b= xa·b.

What happens once you raise an entire expression within parentheses come a power? You have the right to use the approaches you already know to leveling this expression.

(2a)4 = (2a)(2a)(2a)(2a) = (2 • 2 • 2 • 2)(a • a • a • a) = (24)(a4) = 16a4

Notice that the exponent is applied to each element of 2a. So, we can eliminate the middle steps.

(2a)4 = (24)(a4), using the 4 to every factor, 2 and a.

= 16a4

The product of 2 or more numbers raised to a power is same to the product of each number elevated to the exact same power.


A The product of two or an ext non-zero numbers raised to a power equals the product of every number increased to the same power: (ab)x = ax • bx


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For any kind of nonzero numbers a and also b and also any essence x, (ab)x = ax • bx.


Caution! execute not try to apply this dominion to sums. Think around the expression (2 + 3)2. Go (2 + 3)2 same 22 + 32 ? No, it does not—(2 + 3)2 = 52 = 25 and also 22 + 32  = 4 + 9 = 13. So, you deserve to only usage this ascendancy when the numbers inside the parentheses space being multiply (or divided, together we will view next).


Example

Problem

Simplify. (2yz)6

26y6z6

Apply the exponent to every number in the product.

Answer

(2yz)6 = 64y6z6


If the variable has actually an exponent v it, use the power Rule: main point the exponents.


Example

Problem

Simplify. (−7a4b)2

(−7)2(a4)2(b)2

Apply the exponent 2 to each factor within the parentheses.

49a4•2 b2

Square the coefficient and use the Power ascendancy to square (a4)2.

49a8 b2

Simplify.

Answer

(−7a4b)2 = 49a8 b2


Simplify the expression.

(−3x3y2)4

A) x4y4

B) -81xy8

C) 81x12y8

D) 81x7y6


Show/Hide Answer

A) x4y4

Incorrect. Psychic to apply the exponent to all terms within the parentheses, including the coefficient. (−3x3y2)4 = (−3)4(x3)4(y2)4; the correct answer is 81x12y8.

B) -81xy8

Incorrect. Remember to use the exponent to every terms in ~ the parentheses, consisting of the variable x. (−3x3y2)4 = (−3)4(x3)4(y2)4; the exactly answer is 81x12y8.

C) 81x12y8

Correct. ( 3x3y2)4 = (−3)4(x3)4(y2)4 = 81x12y8.

D) 81x7y6

Incorrect. Remember to multiply exponents (not include them) when you are increasing a power to a power. In this case, (−3x3y2)4 = (−3)4(x3)4(y2)4 = −34 · x3·4 · y2·4. The exactly answer is 81x12y8.

A Quotient increased to a Power


Now stop look at what happens if friend raise a quotient come a power. Suppose you have actually

*
 and raise it to the third power.

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You deserve to see that elevating the quotient to the power of 3 can likewise be created as the molecule (3) to the strength of 3, and the denominator (4) come the power of 3.

Similarly, if you space using variables, the quotient raised to a strength is equal to the numerator increased to the power over the denominator elevated to power.

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When a quotient is increased to a power, you can apply the power to the numerator and also denominator individually, as presented below.

*
 =
*

A Quotient increased to a Power

For any kind of number a, any type of non-zero number b, and also any creature x,

*
.


Example

Problem

Simplify.

*

*

Apply the strength to each factor individually.

*

Separate into numerical and variable factors.

*

Simplify by acquisition 2 to the third power and applying the Quotient dominance for exponents—subtract the exponents of matching variables.

*

Simplify. Remember that x0 = 1.

Answer

*
 = 8y3


Simplify the expression.

A)

*

B)

C)

D)


Show/Hide Answer

A) 1

Incorrect. Follow the stimulate of operations. Apply the exponent to every variable in the product first, and also then use the Quotient Rule—subtract the index number in state that have actually the exact same base. The exactly answer is .

B)

Correct. Apply the exponent to every variable in the product and then use the Quotient Rule—subtract the index number in state that have actually the very same base.

C)

Incorrect. Return  is equivalent to , it is no in easiest form. Use the exponent to each variable in the product and also then use the Quotient Rule—subtract the exponents in state that have the very same base. The exactly answer is .

D)

Incorrect. This is an indistinguishable expression however not in most basic form. Usage the Quotient Rule—subtract the index number in terms that have actually the same base. The correct answer is

*

Simplifying and assessing Expressions with Exponents


Now let’s look at some more complicated expressions and see how all the rules can assist make working with exponents easier. While these expressions might look facility at first, simply keep complying with the rules of exponents to simplify them!


Example

Problem

Simplify.

*

*

Recall that with an unfavorable exponents, you deserve to use the reciprocal, and also move it to the denominator through a positive exponent.

*

Separate into numerical and variable factors and apply the exponent to each term.

*

*

Divide 1,024 by 16, and subtract exponents making use of the Quotient Rule.

Answer

*


By making use of the rules of exponents, you’ve now got an equivalent expression, 64x7, i m sorry is much much easier to occupational with 보다

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.

The rules of exponents are likewise helpful when analyzing expressions because that a details value the a variable.


Example

Problem

 Evaluate

*
when x = 4 and y = −2

*

In the denominator, notification that a product is being raised to a power. Usage the rules of index number to leveling the denominator.

*

Simplify 22 and also multiply the exponents of x.

*

Separate into numerical and variable factors.

*

Divide coefficients, usage the Quotient rule to division the variables— subtract the exponents.

*

Simplify. Remember the y0 is 1.

*

Substitute the value 4 for the variable x.

Answer

*
 = 96 once x = 4 and also y = −2


Notice that you might have worked this problem by substituting 4 because that x and 2 because that y in the initial expression. You would still obtain the prize of 96, yet the computation would certainly be much more complex. Notification that you didn’t also need to use the worth of y to advice the above expression.

Simplify.

*

A)

B)

C)

D)


Show/Hide Answer

A)

Incorrect. The very first term has actually been raised to a power, however you have the right to simplify further. The two factors haven"t to be multiplied together. The correct answer is .

B)

Incorrect. The coefficient the the an initial term must likewise be increased to a power. The exactly answer is .

C)

Incorrect. When you take a power of a strength the exponents space multiplied. The exactly answer is .

D)

Correct. In her answer, every variable appears only once, coefficients have actually been multiplied, and there room no strength of powers.

*
, i beg your pardon simplifies come .

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Summary


There are numerous rules to use when working through exponential expressions. You can use these rules, and also the meaning of exponents, to simplify complicated expressions. Simple an expression before analyzing the expression can frequently make the computation easier.