Question
Apply the Quotient Rule of Integer Exponents to simplify 7^4/761. Leave your answer in exponential form.Show your work.
Answers
Answered by
GPT-4o mini
The Quotient Rule of Integer Exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). However, in this case, we're dealing with rational numbers instead of the same base numbers in the exponent form.
Let's express \( 7^4 / 761 \) in its simplest form. Since \( 7^4 \) is a power of 7 and 761 is a constant, we can't directly apply the Quotient Rule because we don't have a common base.
First, we calculate \( 7^4 \):
\[
7^4 = 7 \times 7 \times 7 \times 7 = 49 \times 49 = 2401
\]
Now, we rewrite the expression:
\[
\frac{7^4}{761} = \frac{2401}{761}
\]
Next, we verify if \( 2401 \) can be divided by \( 761 \). Carrying out the division gives us:
\[
\frac{2401}{761} \approx 3.16 \quad (\text{not an integer})
\]
Thus, the result does not simplify nicely to an integer value, and there are no integer factors that we can conveniently use to simplify it further.
Since we are instructed to leave our answer in exponential form and the division doesn't yield a common base exponent, the best representation we can leave is just as is:
\[
\frac{7^4}{761} = \frac{2401}{761}
\]
To express this in a proper form, we may write:
\[
7^4 \times 761^{-1}
\]
Thus, the final simplified representation of your expression is:
\[
7^4 \times 761^{-1}
\]
Let's express \( 7^4 / 761 \) in its simplest form. Since \( 7^4 \) is a power of 7 and 761 is a constant, we can't directly apply the Quotient Rule because we don't have a common base.
First, we calculate \( 7^4 \):
\[
7^4 = 7 \times 7 \times 7 \times 7 = 49 \times 49 = 2401
\]
Now, we rewrite the expression:
\[
\frac{7^4}{761} = \frac{2401}{761}
\]
Next, we verify if \( 2401 \) can be divided by \( 761 \). Carrying out the division gives us:
\[
\frac{2401}{761} \approx 3.16 \quad (\text{not an integer})
\]
Thus, the result does not simplify nicely to an integer value, and there are no integer factors that we can conveniently use to simplify it further.
Since we are instructed to leave our answer in exponential form and the division doesn't yield a common base exponent, the best representation we can leave is just as is:
\[
\frac{7^4}{761} = \frac{2401}{761}
\]
To express this in a proper form, we may write:
\[
7^4 \times 761^{-1}
\]
Thus, the final simplified representation of your expression is:
\[
7^4 \times 761^{-1}
\]