Which expression is correctly developed to use the Power Rule of Exponents?(1 point)

The expression that is correctly developed to use the Power Rule of Exponents is:

\((7^4)^8\)

This expression applies the Power Rule, which states that \((a^m)^n = a^{m \cdot n}\). When applying this rule to the expression \((7^4)^8\), you would multiply the exponents, resulting in \(7^{4 \cdot 8} = 7^{32}\).

The other options do not correctly illustrate the Power Rule of Exponents as clearly.

Which of the following correctly used the Power Rule of Exponents in the expression (109)7
(
10
9
)
7
?(1 point)
Responses

102
10
2
10 superscript 2 baseline

907
90
7
90 superscript 7 baseline

1063
10
63
10 superscript 63 baseline

1016
10
16
10 superscript 16 baseline

The expression \((10^9)^7\) correctly uses the Power Rule of Exponents, which states that \((a^m)^n = a^{m \cdot n}\).

Applying this rule:

\[ (10^9)^7 = 10^{9 \cdot 7} = 10^{63} \]

Thus, the correct response is:

\(10^{63}\) (10 superscript 63).

If the expression 410
4
10
was simplified using the Power Rule of Exponents, what is an equivalent expanded expression? (1 point)
Responses

(45)2
(
4
5
)
2
left parenthesis 4 superscript 5 baseline right parenthesis squared

4⋅4⋅4⋅4⋅4⋅4⋅4
4
⋅
4
⋅
4
⋅
4
⋅
4
⋅
4
⋅
4
4 times 4 times 4 times 4 times 4 times 4 times 4

(42)8
(
4
2
)
8
left parenthesis 4 squared right parenthesis superscript 8 baseline

(412)2

To simplify \(4^{10}\) using the Power Rule of Exponents, we can express it as an equivalent expanded expression.

The simplest expansion of \(4^{10}\) is:

\(4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\) (which is \(4\) multiplied by itself \(10\) times).

So the correct response from your options would be:

\(4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\) (4 times 4 times 4 times 4 times 4 times 4 times 4).

The other options either do not represent \(4^{10}\) correctly or are incorrect forms of the original expression.

Use the Power Rule of Exponents to create an expression equivalent to (156)2
(
15
6
)
2
.(1 point)
Responses

154
15
4
15 superscript 4 baseline

1512
15
12
15 superscript 12 baseline

158
15
8
15 superscript 8 baseline

153

To simplify the expression \((15^6)^2\) using the Power Rule of Exponents, we apply the rule which states that \((a^m)^n = a^{m \cdot n}\).

Here, we have:

\[ (15^6)^2 = 15^{6 \cdot 2} = 15^{12} \]

So, the equivalent expression is:

\(15^{12}\) (15 superscript 12).

Therefore, the correct response is:

1512.

What exponent would make the two expressions equivalent: (33)x=324
(
3
3
)
𝑥
=
3
24
?(1 point)
Responses

8
8

21
21

27
27

15

To find the value of \( x \) that makes the two expressions equivalent, you can follow these steps:

Given the equation:

\[ (3^3)^x = 3^{24} \]

Using the Power Rule of Exponents, we can simplify the left side:

\[ 3^{3x} = 3^{24} \]

Since the bases (3) are the same, we can set the exponents equal to each other:

\[ 3x = 24 \]

Now, solve for \( x \):

\[ x = \frac{24}{3} = 8 \]

Thus, the correct exponent that makes the two expressions equivalent is:

8.