Which expression is correctly developed to use the product rule of exponents?

Question

A)6^2 x 7^3
B) (5^2)^9
C)10^8 x 10^8
D)32^7

Answer 1

To identify which expression can be correctly developed using the product rule of exponents, we first need to understand the product rule. The product rule states that when you multiply two powers with the same base, you can add their exponents:

\[ a^m \times a^n = a^{m+n} \]

Let's evaluate each option:

A) \( 6^2 \times 7^3 \)

• This involves different bases (6 and 7), so the product rule does not apply here.

B) \( (5^2)^9 \)

• This can be simplified using the power rule of exponents (\( (a^m)^n = a^{m \cdot n} \)), not the product rule.

C) \( 10^8 \times 10^8 \)

• This has the same base (10), so we can apply the product rule:
  \[ 10^8 \times 10^8 = 10^{8 + 8} = 10^{16} \]
• This is a correct application of the product rule.

D) \( 32^7 \)

• There is only one base here, so the product rule does not apply.

The expression that is correctly developed to use the product rule of exponents is: C) \( 10^8 \times 10^8 \).

Answer 2

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)200^15
D)100^15

Answer 3

To simplify the expression \(100^8 \times 100^7\) using the Product Rule of Exponents, we apply the rule, which states:

\[ a^m \times a^n = a^{m+n} \]

In this case, both terms have the same base (100). We add the exponents:

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

Now let's evaluate the options given:

A) \(100^{56}\)

• Incorrect, as this does not match our result.

B) \(10,000^{15}\)

• \(10,000 = 100^2\), so this would be \(100^{30}\). Incorrect.

C) \(200^{15}\)

• Incorrect and unrelated to \(100\).

D) \(100^{15}\)

• This is the correct simplification based on our calculation.

The expression that correctly used the Product Rule of Exponents to simplify \(100^8 \times 100^7\) is: D) \(100^{15}\).

Answer 4

Find the numerical equivalent of 10^2 x 10^3
A)1,100
B)10^5
C)10
D)100,000

Answer 5

To find the numerical equivalent of \(10^2 \times 10^3\), we can use the Product Rule of Exponents, which states:

\[ a^m \times a^n = a^{m+n} \]

In this case, we have:

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

Now, let's evaluate the options given:

A) \(1,100\)

• This is incorrect.

B) \(10^5\)

• This is correct as it matches our result.

C) \(10\)

• This is also incorrect.

D) \(100,000\)

• While \(10^5\) is equal to \(100,000\), the answer more directly reflecting our calculation is \(10^5\).

Therefore, the numerical equivalent is B) \(10^5\).

Answer 6

Find the numerical equivalent of 2^5 x 2^-2
A)8
B)16
C)2
D)128

Answer 7

To find the numerical equivalent of \(2^5 \times 2^{-2}\), we can apply the Product Rule of Exponents, which states:

\[ a^m \times a^n = a^{m+n} \]

In this case, we have:

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

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

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

Now let's evaluate the options given:

A) \(8\)

• This is correct.

B) \(16\)

• This is incorrect.

C) \(2\)

• This is incorrect.

D) \(128\)

• This is incorrect.

The numerical equivalent of \(2^5 \times 2^{-2}\) is A) \(8\).

Answer 8

Find the numerical equivalent of 4^9 x 4^-7
A)16
B)4^2
C)4
D)4^16

Answer 9

To find the numerical equivalent of \(4^9 \times 4^{-7}\), we can apply the Product Rule of Exponents, which states:

\[ a^m \times a^n = a^{m+n} \]

In this case, we have:

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

Now let's evaluate the options given:

A) \(16\)

• This is equal to \(4^2\), but we need to express it in the correct form as per the options.

B) \(4^2\)

• This is correct.

C) \(4\)

• This is incorrect.

D) \(4^{16}\)

• This is incorrect.

The numerical equivalent of \(4^9 \times 4^{-7}\) is B) \(4^2\).