QuestionIdentify the expression that applies the Product Rule of Exponents: $6^{2} \cdot 7^{3}$ , $10^{8} \cdot 10^{8}$ , $\left(5^{2}\right)^{9}$ , or $32^{7}$ ?

Studdy Solution

STEP 1

Assumptions1. The Product Rule of Exponents states that for any real numbers a, b, and c, where a is not equal to zero, $(a^b)(a^c) = a^{b+c}$ . . We need to find which of the given expressions correctly applies this rule.

STEP 2

Let's examine the first expression, $6^{2} \cdot7^{}$ .
This expression does not apply the Product Rule of Exponents because the bases (6 and7) are not the same.

STEP 3

Now, let's examine the second expression, $10^{8} \cdot10^{8}$ .
This expression correctly applies the Product Rule of Exponents because the bases are the same (10), and according to the rule, we can add the exponents when the bases are the same.
$10^{8} \cdot10^{8} =10^{8+8}$

STEP 4

Next, let's examine the third expression, $\left(^{2}\right)^{9}$ .
This expression does not apply the Product Rule of Exponents. Instead, it applies the Power of a Power Rule, which states that for any real numbers a, b, and c, where a is not equal to zero, $(a^b)^c = a^{b*c}$ .

STEP 5

Finally, let's examine the fourth expression, $32^{7}$ .
This expression does not apply the Product Rule of Exponents because there is only one base (32) with an exponent.

STEP 6

Based on our analysis, the expression that correctly applies the Product Rule of Exponents is $10^{8} \cdot10^{8}$ .

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QuestionIdentify the expression that applies the Product Rule of Exponents: $6^{2} \cdot 7^{3}$ , $10^{8} \cdot 10^{8}$ , $\left(5^{2}\right)^{9}$ , or $32^{7}$ ?

Studdy Solution

STEP 1

Assumptions1. The Product Rule of Exponents states that for any real numbers a, b, and c, where a is not equal to zero, $(a^b)(a^c) = a^{b+c}$ . . We need to find which of the given expressions correctly applies this rule.

STEP 2

Let's examine the first expression, $6^{2} \cdot7^{}$ .
This expression does not apply the Product Rule of Exponents because the bases (6 and7) are not the same.

STEP 3

Now, let's examine the second expression, $10^{8} \cdot10^{8}$ .
This expression correctly applies the Product Rule of Exponents because the bases are the same (10), and according to the rule, we can add the exponents when the bases are the same.
$10^{8} \cdot10^{8} =10^{8+8}$

STEP 4

Next, let's examine the third expression, $\left(^{2}\right)^{9}$ .
This expression does not apply the Product Rule of Exponents. Instead, it applies the Power of a Power Rule, which states that for any real numbers a, b, and c, where a is not equal to zero, $(a^b)^c = a^{b*c}$ .

STEP 5

Finally, let's examine the fourth expression, $32^{7}$ .
This expression does not apply the Product Rule of Exponents because there is only one base (32) with an exponent.

STEP 6

Based on our analysis, the expression that correctly applies the Product Rule of Exponents is $10^{8} \cdot10^{8}$ .

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QuestionIdentify the expression that applies the Product Rule of Exponents: 62⋅736^{2} \cdot 7^{3}62⋅73, 108⋅10810^{8} \cdot 10^{8}108⋅108, (52)9\left(5^{2}\right)^{9}(52)9, or 32732^{7}327?

Studdy Solution

STEP 1

Assumptions1. The Product Rule of Exponents states that for any real numbers a, b, and c, where a is not equal to zero, (ab)(ac)=ab+c(a^b)(a^c) = a^{b+c}(ab)(ac)=ab+c. . We need to find which of the given expressions correctly applies this rule.

STEP 2

Let's examine the first expression, 62⋅76^{2} \cdot7^{}62⋅7.This expression does not apply the Product Rule of Exponents because the bases (6 and7) are not the same.

STEP 3

STEP 4

STEP 5

Finally, let's examine the fourth expression, 32732^{7}327.This expression does not apply the Product Rule of Exponents because there is only one base (32) with an exponent.

STEP 6

Based on our analysis, the expression that correctly applies the Product Rule of Exponents is 108⋅10810^{8} \cdot10^{8}108⋅108.

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QuestionIdentify the expression that applies the Product Rule of Exponents: $6^{2} \cdot 7^{3}$ , $10^{8} \cdot 10^{8}$ , $\left(5^{2}\right)^{9}$ , or $32^{7}$ ?

Assumptions1. The Product Rule of Exponents states that for any real numbers a, b, and c, where a is not equal to zero, $(a^b)(a^c) = a^{b+c}$ . . We need to find which of the given expressions correctly applies this rule.

Let's examine the first expression, $6^{2} \cdot7^{}$ .
This expression does not apply the Product Rule of Exponents because the bases (6 and7) are not the same.

Finally, let's examine the fourth expression, $32^{7}$ .
This expression does not apply the Product Rule of Exponents because there is only one base (32) with an exponent.

Based on our analysis, the expression that correctly applies the Product Rule of Exponents is $10^{8} \cdot10^{8}$ .