Is the equation 7−exponent4⋅7exponent3/49 1/2=7

Question

true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

True, because the Quotient Property gives a final exponent of 1.
True, because the Quotient Property gives a final exponent of 1.

True, because the Product Property makes the numerator and the denominator equal.
True, because the Product Property makes the numerator and the denominator equal.

False. When applying the Quotient Property, the final exponent is not 1.
False. When applying the Quotient Property, the final exponent is not 1.

False. The answer should be negative since the exponents are negative.

Answer 1

To evaluate the expression \( \frac{7^{-4} \cdot 7^{3}}{49^{1/2}} \) and check if it equals 7, let’s simplify it step by step.

1. Convert \( 49^{1/2} \) into terms of base 7: \[ 49 = 7^2 \implies 49^{1/2} = (7^2)^{1/2} = 7^{2 \cdot \frac{1}{2}} = 7^1 = 7 \]

2. Now rewrite the expression: \[ \frac{7^{-4} \cdot 7^{3}}{7} \]

3. Apply the Product Property of exponents in the numerator: \[ 7^{-4} \cdot 7^{3} = 7^{-4 + 3} = 7^{-1} \]

4. Now simplify the overall expression: \[ \frac{7^{-1}}{7^{1}} = 7^{-1 - 1} = 7^{-2} \]

5. Converting \( 7^{-2} \) back gives: \[ 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \]

Now, we see that \( \frac{1}{49} \) does not equal 7.

Conclusion: The correct response is: False. When applying the Quotient Property, the final exponent is not 1.