Chapter 4.4, Problem 37WE

a)

To determine

To simplify the Sum of given monomials.

Answer to Problem 37WE

The sum of the given monomials is $3a^4{b}^6.$

Explanation of Solution

Given:

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

Calculation: Put plus sign in between given monomials

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

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

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

  $=-a{a}^3{b}^6$

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

  $-a{a}^3{b}^6=-a^4{b}^6$

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

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

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

  $=4a^4{b}^6$

Therefore,

  $a{\left(-a{b}^2\right)}^3+{\left(2a^2{b}^3\right)}^2=-a^4{b}^6+4a^4{b}^6$

Add like terms

  $-a^4{b}^6+4a^4{b}^6=3a^4{b}^6$

Therefore, the sum of the given monomials is $3a^4{b}^6.$

(b)

To determine

To simplify the product of given monomials.

Answer to Problem 37WE

The product of the monomials is $-4a^8{b}^{12}.$

Explanation of Solution

Given:

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

Calculations: Putting product sign in between given monomials

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

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

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

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

  $=-a^3{b}^6$

Now simplify: ${\left(2a^2{b}^3\right)}^2$

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

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

  $=2^2{a}^4{\left(b^3\right)}^2$

  $\Rightarrow 4a^4{b}^6$

Now product: $4a^{4}a{b}^6\left(-a^3{b}^6\right)$

Now remove parenthesis:

  $-4a^4{b}^6\cdot a{a}^3{b}^6$

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

  $-4a^4{b}^6\cdot a{a}^3{b}^6=-b^6\cdot 4a^{1+3+4}{b}^6$

  $=-b^6\cdot 4a^8{b}^6$

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

  $-b^6\cdot 4a^8{b}^6=-4a^8{b}^{6+6}$

  $\Rightarrow -4a^8{b}^{12}$

Therefore, the product of the monomials is $-4a^8{b}^{12}.$