Question

Which expression is correctly developed to use the Product Rule of Exponents?
6^2 times 7^3
32^7
(5^2)^9
10^8 times 10^8

Answer 1

Which of the following correctly used the Product Rule of Exponents to simplify the expression 100^8 times 100^7?
10000^15
100^56
200^15
100^15

Answer 2

Find the numerical equation of 10^2 times 10^3
100000
10
10^5
1100

Answer 3

Find the numerical equivalent of 2^5 times 2^-2
2
128
16
8

Answer 4

Find the numerical equivalent of 4^9 times 4^-7.
4^2
4
4^16
16

Answer 5

The Product Rule of Exponents states that when you multiply two powers that have the same base, you should add their exponents.

Let's analyze the given expressions to see which one correctly utilizes this rule.

1. **\(6^2 \times 7^3\)**: This expression has different bases (6 and 7), so the Product Rule does not apply here.

2. **\(32^7\)**: This is simply a single exponentiation. There is no multiplication of two like bases, so the Product Rule does not apply.

3. **\((5^2)^9\)**: This expression applies the Power Rule of Exponents (which states that \((a^m)^n = a^{m \times n}\)), not the Product Rule.

4. **\(10^8 \times 10^8\)**: This expression has the same base (10). Applying the Product Rule, we would add the exponents:
\[
10^8 \times 10^8 = 10^{8+8} = 10^{16}
\]

Thus, the expression that is correctly developed to use the Product Rule of Exponents is **\(10^8 \times 10^8\)**.

Answer 6

To simplify the expression \(100^8 \times 100^7\) using the Product Rule of Exponents, we apply the rule that states when you multiply two powers with the same base, you add the exponents:

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

Now, let's evaluate the options provided:

1. **\(10000^{15}\)**: This does not match our result.
2. **\(100^{56}\)**: This does not match our result.
3. **\(200^{15}\)**: This does not match our result.
4. **\(100^{15}\)**: This matches our result.

Therefore, the correct simplification of \(100^8 \times 100^7\) using the Product Rule of Exponents is **\(100^{15}\)**.

Answer 7

To solve the expression \(10^2 \times 10^3\) using the Product Rule of Exponents, we add the exponents since the bases are the same:

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

Now, let's evaluate the options provided:

1. **100000**: This is equal to \(10^5\).
2. **10**: This is not the correct answer.
3. **\(10^5\)**: This is the expression we found.
4. **1100**: This is not the correct answer.

So, both **100000** and **\(10^5\)** represent the same value. However, the correct numerical equation representation of \(10^2 \times 10^3\) can be expressed simply as **\(10^5\)** or its numerical value, which is **100000**.

Thus, the answers **100000** and **\(10^5\)** are both correct representations of the product.

Answer 8

To find the numerical equivalent of \(2^5 \times 2^{-2}\), we can use the Product Rule of Exponents. According to this rule, when multiplying powers with the same base, we add the exponents:

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

Now, we can calculate \(2^3\):

\[
2^3 = 8
\]

Now let's evaluate the options provided:

1. **2**: This is not the correct answer.
2. **128**: This is not the correct answer.
3. **16**: This is not the correct answer.
4. **8**: This is the correct answer.

Thus, the numerical equivalent of \(2^5 \times 2^{-2}\) is **8**.

Answer 9

To find the numerical equivalent of \(4^9 \times 4^{-7}\), we can use the Product Rule of Exponents, which states that when multiplying powers with the same base, we add the exponents:

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

Now, we can calculate \(4^2\):

\[
4^2 = 16
\]

Now let’s evaluate the options provided:

1. **\(4^2\)**: This is the correct expression we found (equals 16).
2. **4**: This is not the correct answer.
3. **\(4^{16}\)**: This is not the correct answer.
4. **16**: This is the numerical value of \(4^2\).

Thus, the numerical equivalent of \(4^9 \times 4^{-7}\) is both **\(4^2\)** and **16**. If the question asks for the numeric value, the answer is **16**. If it is looking for the expression, the answer is \(4^2\).