Which expression is correctly developed to use the Product Rule of Exponents?

To apply the Product Rule of Exponents, which states that \( a^m \times a^n = a^{m+n} \), we look for expressions that involve the multiplication of terms with the same base.

Let's analyze the options:

a. \( (5^2)^9 \) - This expression involves the Power Rule (not Product Rule), as it is a power raised to a power.

b. \( 32^7 \) - This is a single term and does not involve multiplication.

c. \( 6^2 \times 7^3 \) - These bases (6 and 7) are different, so this does not apply to the Product Rule.

d. \( 10^8 \times 10^8 \) - This involves the same base, so we can apply the Product Rule here: \( 10^8 \times 10^8 = 10^{8+8} = 10^{16} \).

Thus, the expression that is correctly developed to use the Product Rule of Exponents is:

d. \( 10^8 \times 10^8 \)

Which of the following correctly used the Product Rule of Exponents to simplify the expression 100^8 x 100^7?

a. 100^56
b. 10,000^15
c. 100^15
d. 200^15

To simplify the expression \( 100^8 \times 100^7 \) using the Product Rule of Exponents, we apply the rule which states that \( a^m \times a^n = a^{m+n} \).

In this case:

\[ 100^8 \times 100^7 = 100^{8+7} = 100^{15} \]

Now, let's look at the options provided:

a. \( 100^{56} \) - This is incorrect.

b. \( 10,000^{15} \) - This does not relate to our original calculation directly and does not simplify to the same form.

c. \( 100^{15} \) - This is correct, as we calculated above.

d. \( 200^{15} \) - This is also incorrect.

Thus, the correct answer is:

c. \( 100^{15} \).

find the numerical equivalent of 10^2 x 10^3
a. 10^5
b. 1,100
c. 100,000
d. 10

To find the numerical equivalent of \( 10^2 \times 10^3 \), we apply the Product Rule of Exponents, which states that \( a^m \times a^n = a^{m+n} \).

So,

\[ 10^2 \times 10^3 = 10^{2+3} = 10^5 \]

Now let's look at the options:

a. \( 10^5 \) - This is correct, as it represents the exponentiation result.

b. \( 1,100 \) - This is incorrect.

c. \( 100,000 \) - This is incorrect (though \( 10^5 = 100,000 \), we are looking for the numerical equivalent of the original expression).

d. \( 10 \) - This is incorrect.

The answer is:

a. \( 10^5 \).

Find the numerical equivalent of 2^5 x 2-^2

a. 16
b. 128
c. 8
d. 2

To find the numerical equivalent of \( 2^5 \times 2^{-2} \), we can use the Product Rule of Exponents, which states that \( a^m \times a^n = a^{m+n} \).

Applying this rule, we have:

\[ 2^5 \times 2^{-2} = 2^{5 + (-2)} = 2^{5 - 2} = 2^3 \]

Now, we need to find the numerical equivalent of \( 2^3 \):

\[ 2^3 = 2 \times 2 \times 2 = 8 \]

Now, let's look at the options provided:

a. 16 - This is incorrect.

b. 128 - This is incorrect.

c. 8 - This is correct.

d. 2 - This is incorrect.

Thus, the answer is:

c. 8.

Find the numerical equivalent of 4^9 x 4^-7

a. 16
b. 4^2
c. 4
d. 4^16

To find the numerical equivalent of \( 4^9 \times 4^{-7} \), we can apply the Product Rule of Exponents, which states that \( a^m \times a^n = a^{m+n} \).

Using this rule, we have:

\[ 4^9 \times 4^{-7} = 4^{9 + (-7)} = 4^{9 - 7} = 4^{2} \]

Now, we need to evaluate \( 4^2 \):

\[ 4^2 = 16 \]

Now, let's look at the options provided:

a. 16 - This is correct.

b. \( 4^2 \) - This is also correct because it's the same as 16.

c. 4 - This is incorrect.

d. \( 4^{16} \) - This is incorrect.

Thus, the answer could be expressed in two valid options:

a. 16 and b. \( 4^2 \).

If you are looking for the numerical equivalent, a. 16 is the most straightforward answer.