A) Is the equation (7 ^ - 4 * 7 ^ 3)/(sqrt(49)) = 7 true or false? Select the response that correctly answers the question and provides the appropriate justification.

Question

True, because the Product Property makes the numerator and the denominator equal.
True, because the Quotient Property gives a final exponent of 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 determine whether the equation \((7^{-4} \cdot 7^{3}) / (\sqrt{49}) = 7\) is true or false, we can simplify both sides of the equation step by step.

First, simplify the numerator:

\[ 7^{-4} \cdot 7^{3} \]

Using the Product Property of exponents, which states that \(a^m \cdot a^n = a^{m+n}\):

\[ 7^{-4 + 3} = 7^{-1} \]

Next, simplify the denominator:

\[ \sqrt{49} = 7 \]

Now we have:

\[ \frac{7^{-1}}{7} \]

Using the Quotient Property of exponents, which states that \(\frac{a^m}{a^n} = a^{m - n}\):

\[ \frac{7^{-1}}{7^1} = 7^{-1 - 1} = 7^{-2} \]

Now, we want to know if:

\[ 7^{-2} = 7 \]

This is false because \(7^{-2} = \frac{1}{7^2} = \frac{1}{49}\), which does not equal \(7\).

Therefore, the correct response is:

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