Chapter P.1, Problem 48E

In Exercises 47−52, simplify the expression. Assume that the variables in the denominators are nonzero.

48. $\frac{{\left(3x^2\right)}^2{y}^4}{3y^2}$

Solution

To simplify the expression $\frac{{\left(3x^2\right)}^2{y}^4}{3y^2}$ .

  $3x^4{y}^2$

Given:

The expression, $\frac{{\left(3x^2\right)}^2{y}^4}{3y^2}$ .

Concept Used:

Exponent Properties:

• Product property: $a^m\cdot a^n=a^{m+n}$
• Quotient property: $\frac{a^m}{a^n}=a^{m-n}$
• Power of a power property: ${\left(a^m\right)}^n=a^{m\cdot n}$
• Power to product property: ${\left(ab\right)}^n=a^n\cdot b^n$
• Negative exponent property: $a^{-n}=\frac{1}{a^n}$

Calculation:

In order to solve the given expression $\frac{{\left(3x^2\right)}^2{y}^4}{3y^2}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}\frac{{\left(3x^2\right)}^2{y}^4}{3y^2}=\frac{3^2{x}^{2\cdot 2}{y}^4}{3y^2}\\ =\frac{9x^4{y}^4}{3y^2}\\ =\frac{9}{3}\cdot x^4\cdot \frac{y^4}{y^2}\\ =3\cdot x^4\cdot y^{4-2}\\ =3\cdot x^4\cdot y^2\\ =3x^4{y}^2\end{array}$

Therefore, the simplified expression is $3x^4{y}^2$ .