Chapter 12, Problem 20CLR

To determine

To Evaluate: The simplified value of the expression ${\left(3a^3-2\right)}^2$ .

Answer to Problem 20CLR

The value of the expression ${\left(3a^3-2\right)}^2$ is $9a^6-12a^3+4$ .

Explanation of Solution

Given information:

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

Formula used:

When an algebraic expression multiply by group of algebraic terms connected with addition or subtraction is equal to multiplication of expression with the terms separately. That is $a\left(b\pm c\right)=\left(a\times b\right)\pm \left(a\times c\right)$

If two terms have the same sign, their product is positive.

If two terms have the opposite sign, their product is negative.

We can add terms of an algebraic expression which is like.

Calculation:

Consider the expression is ${\left(3a^3-2\right)}^2$ .

Re writes the expression as,

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

Recall that when an algebraic expression multiply by group of algebraic terms connected with addition or subtraction is equal to multiplication of expression with the terms separately. That is $a\left(b\pm c\right)=\left(a\times b\right)\pm \left(a\times c\right)$

Use it,

  $\begin{array}{c}\left(3a^3-2\right)\left(3a^3-2\right)=\left(\left(3a^3-2\right)\times 3a^3\right)+\left(\left(3a^3-2\right)\times \left(-2\right)\right)\\ =\left(3a^3\times 3a^3\right)+\left(-2\times 3a^3\right)+\left(3a^3\times \left(-2\right)\right)+\left(-2\times \left(-2\right)\right)\end{array}$

Again recall that If two terms have the same sign, their product is positive and

if two terms have the opposite sign, their product is negative.

Apply it,

  $\left(3a^3\times 3a^3\right)+\left(-2\times 3a^3\right)+\left(3a^3\times \left(-2\right)\right)+\left(-2\times \left(-2\right)\right)=9a^6-6a^3-6a^3+4$

Also we know that we can add terms of an algebraic expression which is like.

Again apply it,

  $9a^6-6a^3-6a^3+4=9a^6-12a^3+4$

Thus, the value of the expression ${\left(3a^3-2\right)}^2$ is $9a^6-12a^3+4$ .