Chapter 7, Problem 21MS

To determine

To Solve: Simplify.

Answer to Problem 21MS

The simplified form of expression ${\left(2c^2{d}^{2}e\right)}^6+{\left(-2c^4{d}^4{e}^2\right)}^3$ is $56c^{12}{d}^{12}{e}^6$ .

Explanation of Solution

Given information:

The expression $3a{b}^2\left(a^2-ab+b^2\right)$ .

Formula used:

  $a^m\cdot a^n=a^{m+n}$

Calculation:

  $\begin{array}{l}3a{b}^2\left(a^2-ab+b^2\right)\\ =3a{b}^2\cdot a^2-3a{b}^2\cdot ab+3a{b}^2\cdot b^2\\ =3\left(a\cdot a^2\right)b^2-3\left(a\cdot a\right)\left(b^2\cdot b\right)+3a\left(b^2\cdot b^2\right)\end{array}$

Recall the property of exponents $a^m\cdot a^n=a^{m+n}$ .

  $\begin{array}{l}=3\left(a\cdot a^2\right)b^2-3\left(a\cdot a\right)\left(b^2\cdot b\right)+3a\left(b^2\cdot b^2\right)\\ =3a^{1+3}{b}^2-3a^{1+1}{b}^{2+1}+3a{b}^{2+2}\\ =3a^4{b}^2-3a^2{b}^3+3a{b}^4\end{array}$

The simplified form of expression $3a{b}^2\left(a^2-ab+b^2\right)$ is $56c^{12}{d}^{12}{e}^6$ .