Is the equation 7−4⋅73/49√=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. 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

To determine the truth of the equation \( 7 - \frac{4 \cdot 7^{3/49}}{\sqrt{7}} = 7 \), let's simplify the left side:

First, recall that \( \sqrt{7} = 7^{1/2} \). Hence, we can rewrite the equation as:

\[ 7 - \frac{4 \cdot 7^{3/49}}{7^{1/2}} \]

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

\[ \frac{4 \cdot 7^{3/49}}{7^{1/2}} = 4 \cdot 7^{3/49 - 1/2} \]

Now, convert \( 1/2 \) to a fraction with a denominator of 49: \[ \frac{1}{2} = \frac{49}{98} \]

So we can rewrite it: \[ 1/2 = \frac{24.5}{49} = \frac{48/49}{1} \]

Now, we calculate \( 3/49 - 24.5/49 \): \[ 3/49 - 24.5/49 = \frac{3 - 24.5}{49} = \frac{-21.5}{49} \]

Hence:

\[ 7 - 4 \cdot 7^{-21.5/49} \]

We can now rewrite the expression: \[ 7 - 4 \cdot 7^{-21.5/49} = 7 \]

This indicates that \( 7 - \) some positive quantity (since \( 7^{-21.5/49} \) is a fraction and multiplying by 4 keeps it positive) will NOT be equal to 7.

Therefore, the original statement is false.

The appropriate justification is:

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