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

The Power Rule of Exponents states that \( a^m \times a^n = a^{m+n} \) and \( (a^m)^n = a^{m \cdot n} \). Let's analyze each of the given options to determine which expression can be rewritten using the Power Rule:

a. \((-5)^0\) - This evaluates to 1, according to the rule that any non-zero number raised to the power of 0 is 1. It doesn't involve the power rule in a directly useful way as per your question.

b. \(12^9\) - This is simply a single term with no application of the Power Rule of Exponents involved.

c. \((7^8) \times (7^{10})\) - This can be rewritten using the Power Rule: \[ 7^8 \times 7^{10} = 7^{8 + 10} = 7^{18}. \] This is a correct application of the Power Rule.

d. \((7^4)^8\) - This can also be rewritten using the Power Rule: \[ (7^4)^8 = 7^{4 \cdot 8} = 7^{32}. \] This is also a correct application of the Power Rule.

Both options c and d correctly use the Power Rule of Exponents, so the correct answers are:

c. \((7^8) \times (7^{10})\)
d. \((7^4)^8\)

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

a. 10^16
b. 10^2
c. 90^7
d. 10^63

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

So, in this case:

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

Now, let's evaluate the given options based on this finding:

a. \(10^{16}\) - This is incorrect.

b. \(10^{2}\) - This is incorrect.

c. \(90^{7}\) - This is incorrect.

d. \(10^{63}\) - This is correct.

Therefore, the correct answer is:

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

If the expression 4^10 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression?

a. (4^2)^8
b. 4 x 4 x 4 x 4 x 4 x 4 x 4
c. (4^5)^2
d. (4^12)^2

To find an equivalent expanded expression for \( 4^{10} \) using the Power Rule of Exponents, we can rewrite \( 4^{10} \) in different formats using exponents. Here’s an analysis of the given options:

a. \((4^2)^8\): This can be simplified using the Power Rule: \[ (4^2)^8 = 4^{2 \cdot 8} = 4^{16}. \] This is not equivalent to \( 4^{10} \).

b. \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\): This expression only contains 7 factors of 4. To equal \( 4^{10} \), we need 10 factors, so this is incorrect.

c. \((4^5)^2\): This can also be simplified using the Power Rule: \[ (4^5)^2 = 4^{5 \cdot 2} = 4^{10}. \] This is equivalent to \( 4^{10} \).

d. \((4^{12})^2\): This simplifies to: \[ (4^{12})^2 = 4^{12 \cdot 2} = 4^{24}. \] This is not equivalent to \( 4^{10} \).

From the analysis, the only expression that simplifies correctly to \( 4^{10} \) is:

c. ((4^5)^2.

Use the Power Rule of Exponents to create an expression equivalent to (15^6)^2

a. 15^12
b. 15^3
c. 15^8
d. 15^4

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

Applying this to the given expression:

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

Now let's evaluate the options:

a. \(15^{12}\) - This is correct.

b. \(15^{3}\) - This is incorrect.

c. \(15^{8}\) - This is incorrect.

d. \(15^{4}\) - This is incorrect.

The correct equivalent expression is:

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

What exponent would make the two expression equivalent: (3^3)^x = 3^24?

a. 27
b. 8
c. 15
d. 21

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

Starting with the left side, we apply the Power Rule:

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

Now we set this equal to the right side:

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

Since the bases are the same (both are powers of 3), we can equate the exponents:

\[ 3x = 24. \]

To solve for \( x \), we divide both sides by 3:

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

Thus, the exponent that would make the two expressions equivalent is:

b. 8.