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

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

Let's simplify the equation step by step:

The equation is:

\[ 7 - 4 \cdot 7^3 / \sqrt{49} \]

First, calculate \(\sqrt{49}\):

\[ \sqrt{49} = 7 \]

Now we can rewrite the equation:

\[ 7 - 4 \cdot 7^3 / 7 \]

Next, simplify \(4 \cdot 7^3 / 7\):

Using the Quotient Property of exponents (\(a^m / a^n = a^{m-n}\)), we get:

\[ 4 \cdot 7^3 / 7^1 = 4 \cdot 7^{3-1} = 4 \cdot 7^2 \]

Now substitute this back into the equation:

\[ 7 - 4 \cdot 7^2 \]

Calculate \(4 \cdot 7^2\):

\[ 7^2 = 49, \quad \text{so} \quad 4 \cdot 7^2 = 4 \cdot 49 = 196 \]

Now substitute this back into the equation:

\[ 7 - 196 \]

This simplifies to:

\[ 7 - 196 = -189 \]

Thus, the final expression is \(-189\), which is not equal to \(7\).

Therefore, the correct response is:

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