Let's analyze the equation step-by-step to determine if it is true or false.
We start with the expression:
\[ \frac{7^{-4} \cdot 7^{3}}{\sqrt{49}} \]
Step 1: Simplify the denominator
We know that:
\[ \sqrt{49} = 7 \]
Step 2: Simplify the numerator
Using the product property of exponents (which states that \(a^m \cdot a^n = a^{m+n}\)), we simplify the numerator:
\[ 7^{-4} \cdot 7^{3} = 7^{-4 + 3} = 7^{-1} \]
Step 3: Combine the numerator and the denominator
Now we have:
\[ \frac{7^{-1}}{7} \]
Using the quotient property of exponents (which states that \(\frac{a^m}{a^n} = a^{m-n}\)), we can simplify:
\[ \frac{7^{-1}}{7^1} = 7^{-1 - 1} = 7^{-2} \]
Step 4: Result
The final result \(7^{-2}\) means \(\frac{1}{7^2} = \frac{1}{49}\).
Conclusion
The original equation simplifies to \(7^{-2}\), which is a valid expression.
From the provided options, the correct response is:
True because the quotient property gives a final exponent of -2.
However, since none of the options mention this result, the closest accurate justification is that it simplifies through the properties of exponents correctly and yields a defined value. The assertion that it leads to a final exponent of 1 is incorrect; hence, the best answer considering the options would be:
False when applying the quotient property the final exponent is not 1.