Chapter 7.3, Problem 4P

a.

To determine

To find: The simplified form of the expression ${\left(x^{-2}\right)}^2{\left(3x{y}^5\right)}^4$ .

Answer to Problem 4P

The required simplified form is $81y^{20}$ .

Explanation of Solution

Given information:

The given expression ${\left(x^{-2}\right)}^2{\left(3x{y}^5\right)}^4$ .

Calculation:

  $\begin{array}{l}={\left(x^{-2}\right)}^2{\left(3x{y}^5\right)}^4\\ ={\left(x^{-2}\right)}^2{3}^4{x}^4{\left(y^5\right)}^4\end{array}$

(Raise each factor of $3x{y}^5$ to the third power)

  $=x^{-4}{3}^4{x}^4{y}^{20}$

(Multiply the exponents of a power raised to a power)

  $=3^4{y}^{20}{x}^{-4}{x}^4$

(Commutative property of multiplication)

  $=3^4{y}^{20}{x}^{-4+4}$

(Add the exponents of powers with the same base)

  $\begin{array}{l}=81y^{20}{x}^0\\ =81y^{20}1\\ =81y^{20}\end{array}$

(Simplify)

b.

To determine

To find: The simplified form of the expression ${\left(3c^{\frac{5}{2}}\right)}^4{\left(c^2\right)}^3$ .

Answer to Problem 4P

The required simplified form is $81c^{16}$ .

Explanation of Solution

Given information:

The given expression ${\left(3c^{\frac{5}{2}}\right)}^4{\left(c^2\right)}^3$ .

Calculation:

  $\begin{array}{l}={\left(3c^{\frac{5}{2}}\right)}^4{\left(c^2\right)}^3\\ =3^4{c}^{\frac{5}{2}\cdot 4}{c}^{2\cdot 3}\end{array}$

(Raise each factor of $3c^{\frac{5}{2}}$ to the third power)

  $=3^4{c}^{10}{c}^6$

(Multiply the exponents of a power raised to a power)

  $=3^4{c}^{10}{c}^6$

(Commutative property of multiplication)

  $=3^4{c}^{10+6}$

(Add the exponents of powers with the same base)

  $=81c^{16}$

(Simplify)

c.

To determine

To find: The simplified form of the expression ${\left(6ab\right)}^3{\left(5a^{-3}\right)}^2$ .

Answer to Problem 4P

The required simplified form is $5400b^3{a}^{-3}$ .

Explanation of Solution

Given information:

The given expression ${\left(6ab\right)}^3{\left(5a^{-3}\right)}^2$ .

Calculation:

  $\begin{array}{l}={\left(6ab\right)}^3{\left(5a^{-3}\right)}^2\\ =6^3{a}^3{b}^3{5}^2{\left(a^{-3}\right)}^2\end{array}$

(Raise each factor of $6ab$ and $5a^{-3}$ to the third power)

  $=6^3{a}^3{b}^3{5}^2{a}^{-6}$

(Multiply the exponents of a power raised to a power)

  $=6^3{5}^2{b}^3{a}^3{a}^{-6}$

(Commutative property of multiplication)

  $=6^3{5}^2{b}^3{a}^{3+\left(-6\right)}$

(Add the exponents of powers with the same base)

  $\begin{array}{l}=216\cdot 25b^3{a}^{-3}\\ =5400b^3{a}^{-3}\end{array}$

(Simplify)