The basic of both index number is a, therefore the product ascendancy applies. Include the exponents.

You are watching: A product of repeated factors using a base and exponent

35 · a10

Add the exponents v a usual base.

Answer

5a4 · 7a6 = 35a10


Simplify the expression, keeping the prize in exponential notation.

(4x5)( 2x8)

A) 8x5 • x8

B) 6x13

C) 8x13

D) 8x40


A) 8x5 • x8

Incorrect. 8x5• x8 is tantamount to (4x5)(2x8), yet it quiet is not in easiest form. Simplify x5•x8 by utilizing the Product preeminence to include exponents. The correct answer is 8x13.

B) 6x13

Incorrect. 6x13 is not indistinguishable to (4x5)(2x8). In this incorrect response, the correct exponents were added, but the coefficients were also added together. Castle should have been multiplied. The correct answer is 8x13.

C) 8x13

Correct. 8x13 is identical to (4x5)(2x8). Main point the coefficients (4 • 2) and also apply the Product preeminence to include the index number of the variables (in this instance x) that room the same.

D) 8x40

Incorrect. 8x40  is not tantamount to (4x5)(2x8). Perform not main point the coefficients and the exponents. Remember, utilizing the Product Rule include the exponents once the bases room the same. The correct answer is 8x13.

The Power ascendancy for Exponents


Let’s leveling (52)4. In this case, the basic is 52 and also the exponent is 4, for this reason you multiply 52 4 times: (52)4  = 52 • 52 • 52 • 52 = 58 (using the Product preeminence – add the exponents).

(52)4 is a power of a power. The is the 4th power of 5 come the 2nd power. And we saw over that the prize is 58. Notice that the new exponent is the exact same as the product that the initial exponents: 2 • 4 = 8.

So, (52)4 = 52 • 4  = 58 (which equates to 390,625, if you perform the multiplication).

Likewise, (x4)3 = x4 • 3 = x12.

This leader to an additional rule for exponents—the To advanced a strength to a power, main point the exponents. (xa)b = xa•b


")">Power preeminence for Exponents
. To leveling a power of a power, you multiply the exponents, keeping the base the same. For example, (23)5 = 215.


The Power ascendancy for Exponents

For any kind of positive number x and integers a and b: (xa)b= xa· b.


Example

Problem

Simplify.

6(c4)2

6(c4)2

Since girlfriend are increasing a strength to a power, apply the strength Rule and multiply index number to simplify. The coefficient remains unchanged because it is exterior of the parentheses.

Answer

6(c4)2 = 6c8


Example

Problem

Simplify.

a2(a5)3

*

Raise a5 come the strength of 3 by multiplying the exponents together (the power Rule).

*

Since the exponents share the same base, a, they can be linked (the Product Rule).

*

Answer

*


Simplify:

*

A)

B)

C)

D)


Show/Hide Answer

A)

Incorrect. This expression is not streamlined yet. Recall the –a can also be composed –a1. Multiply –a1 through a8 to come at the exactly answer. The exactly answer is .

B)

Incorrect. Execute not add the exponents of 2 and also 4 together. The Power preeminence states that for a power of a power you main point the exponents. The exactly answer is .

C)

Incorrect. Carry out not include the exponents of 2 and also 4 together. The Power dominance states the for a power of a power you main point the exponents. The correct answer is .

D)

Correct. Utilizing the strength Rule,

*
.

The Quotient ascendancy for Exponents


Let’s look at at dividing terms comprise exponential expressions. What happens if girlfriend divide 2 numbers in exponential kind with the very same base? take into consideration the following expression.

You can rewrite the expression as:

*
. Then you have the right to cancel the common factors the 4 in the numerator and also denominator:
*

Finally, this expression have the right to be rewritten as 43 using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and also 2.

So,  = 45-2 = 43.

Be cautious that girlfriend subtract the exponent in the denominator indigenous the exponent in the numerator.

*

or

*
= x7−9 = x-2

So, to divide two exponential terms through the same base, subtract the exponents.


The For any type of non-zero number x and any integers a and also b:

*


")">Quotient preeminence for Exponents

For any kind of non-zero number x and also any integers a and b:

*


Notice that  = 40. And we recognize that  =

*
 = 1. Therefore this may assist to define why 40 = 1.


Example

Problem

Evaluate.

*

*

These two exponents have actually the very same base, 4. Follow to the Quotient Rule, you have the right to subtract the power in the denominator native the power in the numerator.

answer

*
 = 45


When dividing terms that additionally contain coefficients, division the coefficients and then divide variable powers with the exact same base by subtracting the exponents.


Example

Problem

Simplify.

*

*

Separate right into numerical and variable factors.

*

Since the bases the the exponents space the same, girlfriend can apply the Quotient Rule. Divide the coefficients and subtract the exponents of corresponding variables.

Answer

*
=
*


Applying the Rules


All of this rules that exponents—the Product Rule, the power Rule, and also the Quotient Rule—are advantageous when analyzing expressions with usual bases.


Example

Problem

Evaluate

*
 when x = 4.

*

Separate into numerical and also variable factors.

*

Divide coefficients, and also subtract the index number of the variables.

*

Simplify.

*

Substitute the worth 4 because that the change x.

Answer

*
 = 768


Usually, the is less complicated to simplify the expression prior to substituting any values for your variables, however you will get the exact same answer either way.


Example

Problem

Simplify.

*
 

*

Use the order of operations with PEMDAS:

E: advice exponents. Use the Power dominion to leveling (a5)3.

*

*

M: Multiply, utilizing the Product ascendancy as the bases space the same.

*

*

D: Divide using the Quotient Rule.

See more: Wh A Pleasure To Make Your Acquaintance Meaning, Make Somebody'S Acquaintance

price

*
 =
*


Summary


There room rules that assist when multiplying and dividing exponential expressions through the exact same base. Come multiply 2 exponential terms through the very same base, include their exponents. To raise a strength to a power, main point the exponents. To divide 2 exponential terms through the same base, subtract the exponents.