Is the equation 7^−4⋅7^3/√49=7 true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)

Question

Responses

False. The answer should be negative since the exponents are negative.
False. The answer should be negative since the exponents are negative.
False. When applying the Quotient Property, the final exponent is not 1.
False. When applying the Quotient Property, the final exponent is not 1.
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.

Answer 1

To determine whether the equation \( 7^{-4} \cdot 7^3 / \sqrt{49} = 7 \) is true or false, let's simplify the left side step by step.

1. First, simplify \( \sqrt{49} \): \[ \sqrt{49} = 7 \]

2. Substitute this back into the expression: \[ 7^{-4} \cdot 7^3 / 7 \]

3. Now apply the Product Property of exponents, which states that \( a^m \cdot a^n = a^{m+n} \): \[ 7^{-4} \cdot 7^3 = 7^{-4+3} = 7^{-1} \]

4. Now, substitute \( 7^{-1} \) back into the expression: \[ 7^{-1} / 7 \]

5. Using the Quotient Property of exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \): \[ 7^{-1} / 7 = 7^{-1-1} = 7^{-2} \]

6. The result is \( 7^{-2} \), which is not equal to \( 7 \).

Therefore, the equation \( 7^{-4} \cdot 7^3 / \sqrt{49} = 7 \) is false.

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