Is the equation 7−4⋅73/49√=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.

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

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 Product Property makes the numerator and the denominator equal.

Answer 1

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

1. Calculate \( \sqrt{49} \): \[ \sqrt{49} = 7 \]

2. Rewrite the equation: \[ 7 - 4 \cdot \frac{7^3}{7} \]

3. Using the Quotient Property of exponents: \[ \frac{7^3}{7} = 7^{3-1} = 7^2 \] Thus, we have: \[ 7 - 4 \cdot 7^2 \]

4. Calculate \( 7^2 \): \[ 7^2 = 49 \]

5. Substitute back into the equation: \[ 7 - 4 \cdot 49 \]

6. Calculate \( 4 \cdot 49 \): \[ 4 \cdot 49 = 196 \]

7. Final calculation: \[ 7 - 196 = -189 \]

Now, we compare the left side, which is \(-189\), to the right side, which is \(7\). Clearly, \(-189 \neq 7\).

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

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