Questions: Simplify [ left(x^frac12 x^2 y^frac43right)^frac32 ]

Solution Steps

To simplify the given expression, apply the power of a power property \((a^m)^n = a^{m \cdot n}\) to each component inside the parentheses. Then, multiply the exponents of like bases.

We start with the expression: \[ \left(x^{\frac{1}{2}} x^{2} y^{\frac{4}{3}}\right)^{\frac{3}{2}} \]

Using the property \((a^m)^n = a^{m \cdot n}\), we can rewrite the expression as: \[ \left(x^{\frac{1}{2} + 2} y^{\frac{4}{3}}\right)^{\frac{3}{2}} \] Calculating the exponent for \(x\): \[ \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \] Thus, the expression becomes: \[ \left(x^{\frac{5}{2}} y^{\frac{4}{3}}\right)^{\frac{3}{2}} \]

Now, applying the power of a power property again: \[ x^{\frac{5}{2} \cdot \frac{3}{2}} y^{\frac{4}{3} \cdot \frac{3}{2}} = x^{\frac{15}{4}} y^{2} \]

Final Answer