Question

enter the correct answer in the box. simplify the expression $x^{-\frac{4}{5}} \cdot x^{\frac{7}{8}}$.

Explanation:

Step1: Recall exponent rule for multiplication

When multiplying two exponential expressions with the same base \(a\), we use the rule \(a^m \cdot a^n = a^{m + n}\). Here, the base is \(x\), so we add the exponents.
The exponents are \(-\frac{4}{5}\) and \(\frac{7}{8}\). So we need to calculate \(-\frac{4}{5}+\frac{7}{8}\).

Step2: Find a common denominator

The common denominator of 5 and 8 is 40. Convert the fractions:
\(-\frac{4}{5}=-\frac{4\times8}{5\times8}=-\frac{32}{40}\)
\(\frac{7}{8}=\frac{7\times5}{8\times5}=\frac{35}{40}\)

Step3: Add the fractions

Now add the two fractions: \(-\frac{32}{40}+\frac{35}{40}=\frac{- 32 + 35}{40}=\frac{3}{40}\)

Step4: Write the simplified expression

Using the exponent rule, \(x^{-\frac{4}{5}}\cdot x^{\frac{7}{8}}=x^{-\frac{4}{5}+\frac{7}{8}} = x^{\frac{3}{40}}\)

Answer:

\(x^{\frac{3}{40}}\)