Question

an equation is shown below.
\\(\frac{7^{\frac{3}{4}}}{7^{\frac{x}{8}}}=\sqrt8{7}\\)
what value of x makes this equation true?
a. \\(-58\\)
b. \\(5\\)
c. \\(6\\)
d. \\(48\\)

Explanation:

Step1: Use exponent rule for division

When dividing exponents with the same base, we subtract the exponents: \( \frac{a^m}{a^n}=a^{m - n} \). So for \( \frac{7^{\frac{3}{4}}}{7^{\frac{x}{8}}} \), this becomes \( 7^{\frac{3}{4}-\frac{x}{8}} \).

Step2: Rewrite the right - hand side as an exponent

The \( n\)th root of a number \( a\) can be written as \( a^{\frac{1}{n}} \). So \( \sqrt[8]{7}=7^{\frac{1}{8}} \).

Step3: Set the exponents equal

Since the bases are the same (both are 7) and the two expressions are equal, their exponents must be equal. So we have the equation:
\( \frac{3}{4}-\frac{x}{8}=\frac{1}{8} \)

Step4: Solve for x

First, multiply through by 8 to clear the fractions:
\( 8\times\frac{3}{4}-8\times\frac{x}{8}=8\times\frac{1}{8} \)
\( 6 - x = 1 \)
Then, subtract 6 from both sides:
\( -x=1 - 6=-5 \)
Multiply both sides by - 1:
\( x = 5 \)

Answer:

B. 5