Simplify \(\left(\frac{64 x^6}{y^3}\right)^{\frac{2}{3}}\).
Step 1: Apply the fractional exponent to both the numerator and denominator
Apply the rule: \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} \).
Thus: \(\left(\frac{64x^6}{y^3}\right)^{\frac{2}{3}} = \frac{\left(64x^6\right)^{\frac{2}{3}}}{\left(y^3\right)^{\frac{2}{3}}}\).
Step 2: Simplify the numerator \(\left(64x^6\right)^{\frac{2}{3}}\).
Apply the rule: \( (ab)^m = a^mb^m \)
- Split into two parts: \(64^{\frac{2}{3}} \) and \((x^6)^{\frac{2}{3}}\).
Part 1: Simplify \(64^{\frac{2}{3}}\).
- Write 64 as a power of 4: \(64 = 4^3\).
- Then, \(64^{\frac{2}{3}} = \left(4^3\right)^{\frac{2}{3}} = 4^{2} = 16 \).
Part 2: Simplify \((x^6)^{\frac{2}{3}}\).
- Use the rule\((x^m)^n = x^{m \cdot n}\)
- \((x^6)^{\frac{2}{3}} = x^{6 \cdot \frac{2}{3}} = x^4\)
Thus, the numerator becomes: \(\left(64x^6\right)^{\frac{2}{3}} = 16x^4 \).
Step 3: Simplify the denominator \((y^3)^{\frac{2}{3}} \).
- Use the rule \((x^m)^n = x^{m \cdot n}\)
- \((y^3)^{\frac{2}{3}} = y^{3 \cdot \frac{2}{3}} = y^2\).
Step 4: Combine the results.
The simplified expression is: \(\frac{\left(64x^6\right)^{\frac{2}{3}}}{\left(y^3\right)^{\frac{2}{3}}} = \frac{16x^4}{y^2}\).
Final Answer:
\({\frac{16x^4}{y^2}}\)
Laws of Indices
$$
\begin{gathered}
a^m \times a^n=a^{m+n} \\
a^m \div a^n=a^{m-n} \\
\left(a^m\right)^n=a^{mn} \\
a^0=1 \\
a^{-m}=\frac{1}{a^m} \\
a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m
\end{gathered}
$$
