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

Question

(-5)^0
(7^4)^8
12^9
(7^8) times (7^10)

Answer 1

The Power Rule of Exponents states that \( (a^m)^n = a^{m \cdot n} \). The expression that correctly develops to use the Power Rule of Exponents among the options provided is:

\((7^4)^8\)

This expression can be simplified using the Power Rule:

\[ (7^4)^8 = 7^{4 \cdot 8} = 7^{32} \]

The other expressions either do not involve a power raised to another power or are already in their simplest form according to the Power Rule.

Answer 2

Which of the following correctly used the Power Rule of Exponents in the expression (10^9)^7?
10^63
10^16
90^7
10^2

Answer 3

To apply the Power Rule of Exponents to the expression \( (10^9)^7 \), we use the rule that states \( (a^m)^n = a^{m \cdot n} \).

Applying this rule:

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

Therefore, the correct expression that used the Power Rule of Exponents is:

\(10^{63}\).

Answer 4

If the expression 4^10 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression?
4 times 4 times 4 times 4 times 4 times 4 times 4.
(4^2)^8
(4^12)^2
(4^5)^2

Answer 5

The expression \(4^{10}\) can be expanded as the product of \(4\) multiplied by itself 10 times, which is:

\[ 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \]

In terms of providing an equivalent expanded expression using the Power Rule of Exponents, we need to find another way to represent \(4^{10}\).

Now let’s check the options:

1. \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\) - This is not the correct representation, as it only shows \(7\) multiplications.

2. \((4^2)^8\) - By applying the Power Rule: \[ (4^2)^8 = 4^{2 \cdot 8} = 4^{16} \quad \text{(not equal to \(4^{10}\))} \]

3. \((4^{12})^2\) - By applying the Power Rule: \[ (4^{12})^2 = 4^{12 \cdot 2} = 4^{24} \quad \text{(not equal to \(4^{10}\))} \]

4. \((4^5)^2\) - By applying the Power Rule: \[ (4^5)^2 = 4^{5 \cdot 2} = 4^{10} \quad \text{(this is equal to \(4^{10}\))} \]

The only option that correctly represents an equivalent expression for \(4^{10}\) using the Power Rule of Exponents is:

\((4^5)^2\).

Answer 6

Use the Power Rule of Exponents to create an expression equivalent to (15^6)^2.
15^12
15^4
15^3
15^8

Answer 7

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

So we have:

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

Therefore, the expression equivalent to \((15^6)^2\) is:

\(15^{12}\).

Answer 8

What exponent would make the two expressions equivalent: (3^3)^x = 3^24?
15
21
8
27

Answer 9

To find the exponent \(x\) that makes the expressions \((3^3)^x\) and \(3^{24}\) equivalent, we can use the Power Rule of Exponents.

Starting with the left side:

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

Now we set the exponents equal to each other since the bases are the same:

\[ 3x = 24 \]

Next, we solve for \(x\):

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

Thus, the exponent \(x\) that would make the two expressions equivalent is:

8.