Chapter 7.8, Problem 58HP

(a)

To determine

To find: The product of the given expression.

Answer to Problem 58HP

The product is $a^3+3a^{2}b+3a{b}^2+b^3$ .

Explanation of Solution

Given information:

The given expression is ${\left(a+b\right)}^3$

Calculation:

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

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

Therefore, the product is $a^3+3a^{2}b+3a{b}^2+b^3$ .

(b)

To determine

To find: The value of the given expression.

Answer to Problem 58HP

The area is $a^2-b^2.$

Explanation of Solution

Given information:

The given expression is ${\left(x+2\right)}^3$ .

Calculation:

Let $a=x$ and $b=2$

  $\begin{array}{c}{\left(a+b\right)}^3=a^3+3a^{2}b+3a{b}^2+b^3\\ {\left(x+2\right)}^3={\left(x\right)}^3+3{\left(x\right)}^2\left(2\right)+3\left(x\right){\left(2\right)}^2+{\left(2\right)}^3\\ =x^3+6x^2+12x+8\end{array}$

Therefore, the value of the expression is $x^3+6x^2+12x+8$ .

(c)

To determine

To draw: The cube according to the given information.

Answer to Problem 58HP

The cube is shown in Figure-(1).

Explanation of Solution

Given information:

The given side is $\left(a+b\right)$ in length.

Calculation:

The cube according to the given information is shown in figure $\left(1\right)$ .

  

Figure- $\left(1\right)$ .

Therefore, the cube is shown in Figure-(1).

(d)

To determine

To find: The pattern for the cube in given difference.

Answer to Problem 58HP

The pattern is $a^3-3a^{2}b+3a{b}^2-b^3$ .

Explanation of Solution

Given information:

The given difference is ${\left(a-b\right)}^3$ .

Calculation:

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

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

Therefore, the pattern is $a^3-3a^{2}b+3a{b}^2-b^3$ .