Chapter 4.4, Problem 34WE

a)

To determine

To simplify the Sum of given monomials.

Answer to Problem 34WE

The sum of the given monomials is $-7a^{12}.$

Explanation of Solution

Given:

We have given two monomials: ${\left(a^2\right)}^6,{\left(-2a^4\right)}^3$

Calculation:

We have monomials: ${\left(a^2\right)}^6\text{and}{\left(-2a^4\right)}^3.$

Simplify: ${\left(a^2\right)}^6$

Apply exponent rule: ${\left(x^m\right)}^n=x^{m\times n}$

  $\Rightarrow a^{2\times 6}$

  $\Rightarrow a^{12}$

Simplify: ${\left(-2a^4\right)}^3$

Apply exponent rule: ${\left(a\cdot b\right)}^n=a^n{b}^n$

  ${\left(-2\right)}^3{\left(a^4\right)}^3$

  $\Rightarrow {\left(-2\right)}^3{a}^{12}$ $\left[\therefore {\left(-2\right)}^3=-8,{\left(a^4\right)}^3=a^{12}\right]$

  $\Rightarrow -8a^{12}$

Now add both: $a^{12}$ and $-8a^{12}$

  $a^{12}+\left(-8a^{12}\right)$

  $\Rightarrow a^{12}-8a^{12}$

Subtract like terms

  $-7a^{12}$

Therefore, the sum of the given monomials is $-7a^{12}.$

(b)

To determine

The product of given monomials

Answer to Problem 34WE

The product of the monomials is $-8x^{24}.$

Explanation of Solution

Given:

We have given two monomials: ${\left(a^2\right)}^6,{\left(-2a^4\right)}^3$

Calculation:

Now we need to find the product of

  ${\left(a^2\right)}^6,{\left(-2a^4\right)}^3$

Simplify: ${\left(a^2\right)}^6$

Apply exponent rule: ${\left(x^m\right)}^n=x^{m\times n}$

  $\Rightarrow a^{2\times 6}$

  $\Rightarrow a^{12}$

Simplify: ${\left(2x^2\right)}^3$

Apply exponent rule: ${\left(ab\right)}^c=a^c{b}^c$

  $\Rightarrow {\left(-2\right)}^3{\left(a^4\right)}^3$

Apply exponent rule: ${\left(x^m\right)}^n=x^{m\times n}$

  $\Rightarrow {\left(-2\right)}^3{a}^{4\times 3}$

  $\Rightarrow {\left(-2\right)}^3{a}^{12}$

  $\left[\therefore {\left(-2\right)}^3=-8\right]$

  $\Rightarrow -8a^{12}$

Now product the monomials:

  $\Rightarrow \Rightarrow a^{12}\cdot \left(-8a^{12}\right)$

Apply exponent rule: $a^b\cdot a^c=a^{b+c}$

  $\Rightarrow \left[1\cdot \left(-8\right)\right]a^{12+12}$

  $\Rightarrow -8a^{24}$

Therefore, the product of the monomials is $-8x^{24}.$