Chapter 7.3, Problem 27PPE

To determine

To simplify: The expression ${\left(2y^{\frac{7}{9}}\right)}^{-3}$ .

Answer to Problem 27PPE

The required answer is $\frac{1}{8y^{\frac{7}{3}}}$ .

Explanation of Solution

Given information:

The given expression ${\left(2y^{\frac{7}{9}}\right)}^{-3}$ .

Calculation:

  $\begin{array}{l}={\left(2y^{\frac{7}{9}}\right)}^{-3}\\ =2^{-3}{\left(y^{\frac{7}{9}}\right)}^{-3}\end{array}$

(Raise each factor to the second power)

  $\begin{array}{l}=2^{-3}{y}^{\frac{7}{9}\cdot \left(-3\right)}\\ =2^{-3}{y}^{-\frac{7}{3}}\end{array}$

(Multiply the exponents of a power raised to a power)

  $=\frac{1}{8}\cdot \frac{1}{y^{\frac{7}{3}}}$

(The expression using only positive exponents)

  $=\frac{1}{8y^{\frac{7}{3}}}$

(Simplify)