Which expression is correctly developed to use the Power Rule of Exponents? 12^9 (7^4)^8 (−5)^0 (7^8)⋅(7^10)

Answer 1

To use the Power Rule of Exponents effectively, we need to identify which of the expressions can be simplified using the Power Rule. The Power Rule states that \( (a^m)^n = a^{m \cdot n} \), and for multiplication, it states that \( a^m \cdot a^n = a^{m+n} \).

Let's analyze the given expressions:

1. \( 12^9 \) - This expression is already in the simplest form and doesn't invoke the Power Rule.

2. \( (7^4)^8 \) - This can be simplified using the Power Rule: \[ (7^4)^8 = 7^{4 \cdot 8} = 7^{32}. \]

3. \( (−5)^0 \) - This expression evaluates to 1, since any non-zero number raised to the power of 0 is 1. It does not involve using the Power Rule as stated.

4. \( (7^8) \cdot (7^{10}) \) - This can also be simplified using the multiplication rule: \[ (7^8) \cdot (7^{10}) = 7^{8+10} = 7^{18}. \]

Thus, the expressions that are correctly developed using the Power Rule are \( (7^4)^8 \) and \( (7^8) \cdot (7^{10}) \).

If you're looking for the expression that explicitly shows the use of the Power Rule, then \( (7^4)^8 \) is a direct application of the rule \( (a^m)^n \). On the other hand, \( (7^8) \cdot (7^{10}) \) exemplifies the product rule for exponents. Therefore, both expressions accurately apply exponent rules, but in the context of the Power Rule specifically, \( (7^4)^8 \) is the clearest instance.

Answer 2

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Answer 3

Sure! Based on the expressions provided, the one that correctly demonstrates the use of the Power Rule of Exponents is:

(7^4)^8

This expression can be simplified using the Power Rule:

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

Answer 4

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Answer 5

Thank you so much for your kind words! I'm glad I could help. If you have any more questions or need assistance with anything else, feel free to ask!

Answer 6

What exponent would make the two expressions equivalent: (3^3)^x=3^24 ?(1 point) Responses 15 8 21 27

Answer 7

To find the value of \( x \) that makes the two expressions equivalent, we can use the Power Rule of Exponents:

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

Setting this equal to the right side of the equation:

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

Since the bases 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 value of \( x \) that makes the expressions equivalent is 8.