Question

Simplify the given expression. Assume that no denominator equals zero.

${\left(\frac{3x{y}^3}{2z}\right)}^3$

Short answer

The final expression is $\frac{216x^3{y}^9}{z^3}$.

Step-by-step solution

Step 1. Define an expression.

A mathematical expression is a combinations of numbers and variables ordered with mathematical operations.

Step 2. Define power of a power rule.

The power of a power rule states that if a base raised to a power is being raised to another power, the exponents are multiplied and the base remains the same.

${\left(x^m\right)}^n=x^{mn}$

Where,$x$ is any non-zero real number.

Step 3. Simplify the expression 3xy32z3.

Use the property of power of a power rule.

$$\begin{gathered} {\left(\frac{12x{y}^3}{2z}\right)}^3=\frac{{\left(12x{y}^3\right)}^3}{{\left(2z\right)}^3} \\ =\frac{{\left(12\right)}^3{\left(x\right)}^3{\left(y^3\right)}^3}{{\left(2\right)}^3{\left(z\right)}^3} \\ =\frac{1728\left(x^3\right)\left(y^{3\times 3}\right)}{8z^3} \\ =\frac{216x^3{y}^9}{z^3} \end{gathered}$$

Hence,${\left(\frac{12x{y}^3}{2z}\right)}^3=\frac{216x^3{y}^9}{z^3}$

Therefore the final expression is $\frac{216x^3{y}^9}{z^3}$.