Chapter 11.7, Problem 11WE

To determine

To calculate:

Expansion and simplification of expression

  ${\left(p^2+q\right)}^3$

Answer to Problem 11WE

The simplified expression is $p^6+3p^4{q}^3+3p^2{q}^6+q^9$

Explanation of Solution

Given information:

The given expression is,

  ${\left(p^2+q\right)}^3$

Formula:

The formula of perfect cube is,

  ${\left(a+b\right)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

Calculation:

The given expression is,

  ${\left(p^2+q\right)}^3$

Here, consider

  $\begin{array}{l}a=p^2\\ b=q^3\end{array}$

The formula of cube is,

  ${\left(a+b\right)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

Substitute the values in the above formula,

  ${\left(p^2\right)}^3+3{\left(p^2\right)}^2{q}^3+3p^2{\left(q^3\right)}^2+{\left(q^3\right)}^3$

Simplifying the above expression,

In the first step simplify the first bracket ${\left(p^2\right)}^3$

  $\begin{array}{l}{\left(p^2\right)}^3=p^{2\cdot 3}\left[\text{Apply}\text{exponential}\text{rule}{\left(a^b\right)}^c=a^{bc}\right]\\ =p^6\left[\text{Mupltiply}\text{the}\text{two}\text{numbers}\text{2}\times \text{3=6}\right]\end{array}$

Now, simplify the second bracket $3{\left(p^2\right)}^2{q}^3$ ,

  $\begin{array}{l}=3{\left(p^2\right)}^2{q}^3\\ ={\left(p^2\right)}^2=p^4\\ =3p^4{q}^3\end{array}$

Simplify the third bracket $3p^2{\left(q^3\right)}^2$ ,

  $\begin{array}{l}=3p^2{\left(q^3\right)}^2\\ ={\left(q^3\right)}^2=q^6\\ =3p^2{q}^6\end{array}$

Simplify the forth bracket ${\left(q^3\right)}^3$ ,

  $\begin{array}{l}={\left(q^3\right)}^3=q^9\left[\text{Mupltiply}\text{the}\text{numbers}3\cdot 3=9\right]\\ =3p^2{q}^6\end{array}$

Now, the combined expression is written as,

  $p^6+3p^4{q}^3+3p^2{q}^6+q^9$