Chapter 0, Problem 1P

Use the exponent rules to simplify the following expressions:

1. (−3x⁴y²)²

To determine

Use the exponent rules to simply the expression ${\left(-3x^4{y}^2\right)}^2$.

Answer to Problem 1P

The simplified expression is $9x^8{y}^4$.

Explanation of Solution

The product rule: ${\left(x\right)}^m{\left(x\right)}^n=x^{m+n}$

The quotient rule: $\frac{x^m}{x_n}=x^{m-n}$

The first power rule: ${\left(x^m\right)}^n=x^{mn}$

Other power rules:

$\begin{array}{c}{\left(xy\right)}^m=\left(x^m\right)\left(y^m\right)\\ {\left(\frac{x}{y}\right)}^m=\frac{x^m}{y^m}\end{array}$

Conclusion:

Apply the power rule.

$\begin{array}{c}{\left(-3x^4{y}^2\right)}^2={\left(-3\right)}^2{\left(x^4\right)}^2{\left(y^2\right)}^2\\ =\left(9\right)\left(x^{4\times 2}\right)\left(y^{2\times 2}\right)\\ =9x^8{y}^4\end{array}$