Is the equation 7^−4⋅7^3/49√=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. 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.

Answer 1

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

First, we can rewrite \( 49 \) as \( 7^2 \) (since \( 49 = 7 \times 7 \)) and remember that \( \sqrt{} \) indicates the square root. Thus, \( 49\sqrt{} = 7^2 \) can be handled as follows:

1. Convert \( 49 \) to \( 7^2 \): \[ 49 = 7^2 \] So, we can replace \( 49 \) in our expression.

Next, the left-hand side of the equation becomes: \[ \frac{7^{-4} \cdot 7^{3}}{7^2} \]

Now apply the Product Property of exponents to simplify \( 7^{-4} \cdot 7^{3} \): \[ 7^{-4 + 3} = 7^{-1} \]

Now substitute this back into our equation: \[ \frac{7^{-1}}{7^2} \]

Next, apply the Quotient Property of exponents: \[ 7^{-1 - 2} = 7^{-3} \]

Thus, we have: \[ 7^{-3} \]

Now compare this to the right-hand side of the original equation which is \( 7 \) (or \( 7^{1} \)): \[ 7^{-3} \neq 7^{1} \]

Since \( 7^{-3} \) is not equal to \( 7 \), the equation is false.

Based on this analysis, the best answer is: False. When applying the Quotient Property, the final exponent is not 1.