Q. 5.42 - Intermediate Algebra

Question

Simplify each expression:

$ⓐ {\left(a^3{b}^2\right)}^6{\left(4a{b}^3\right)}^4 ⓑ \frac{\left(p-3\right)4( p5)3}{\left(p7\right)6} ⓒ {\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}$

Step-by-Step Solution

Answer

The simplified expression of ${\left(a^3{b}^2\right)}^6{\left(4a{b}^3\right)}^4=256a^{22}{b}^{24}$

The simplified expression of $\frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6}=\frac{1}{p^{39}}$

The simplified expression of ${\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}=2x^3{y}^{10}$

Step 1. Given Information

In the given question we have to simplify each expression: 

$ⓐ {\left(a^3{b}^2\right)}^6{\left(4a{b}^3\right)}^4 ⓑ \frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6} ⓒ {\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}$

Part (a) Step 1. The given expression is ( a 3 b 2 ) 6 ( 4 a b 3 ) 4

Use the Product to a Power Property, ${\left(ab\right)}^m=a^m{b}^m$

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

Simplify.

${\left(a^3{b}^2\right)}^6{\left(4a{b}^3\right)}^4=\left(a^{3·6}\right)\left(b^{2·6}\right)\left(256\right){\left(a\right)}^4\left(b^{3·4}\right)$

${\left(a^3{b}^2\right)}^6{\left(4a{b}^3\right)}^4=\left(a^{18}\right)\left(b^{12}\right)\left(256\right){\left(a\right)}^4\left(b^{12}\right)$

Use the Commutative Property.

${\left(a^3{b}^2\right)}^6{\left(4a{b}^3\right)}^4=256{\left(a\right)}^{4+18}\left(b^{12+12}\right)$

Multiply the constants and add the exponents.

${\left(a^3{b}^2\right)}^6{\left(4a{b}^3\right)}^4=256a^{22}{b}^{24}$

Part (b) Step 1. The given expression is ( p − 3 ) 4 (   p 5 ) 3 ( p 7 ) 6

Use the Power Property, ${\left(a^m\right)}^n=a^{m·n}$

$\frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6}=\frac{\left(p^{-3·4}\right)( p^{5·3})}{\left(p^{7·6}\right)}$

$\frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6}=\frac{\left(p^{-12}\right)( p^{15})}{\left(p^{42}\right)}$

Add the exponents in the numerator.

$\frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6}=\frac{( p^{15-12})}{\left(p^{42}\right)}$

$\frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6}=\frac{( p^3)}{\left(p^{42}\right)}$

Use the Quotient Property $\frac{a^m}{a^n}=\frac{1}{a^{n-m}}$

$\frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6}=\frac{1}{\left(p^{42-3}\right)}$

$\frac{{\left(p^{-3}\right)}^4{( p^5)}^3}{{\left(p^7\right)}^6}=\frac{1}{p^{39}}$

Part (c) Step 1. The given expression is 4 x 3 y 2 x 2 y − 1 2   8 x y − 3 x 2 y − 1

Simplify inside the parentheses first.

${\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}={\left(4x{y}^3\right)}^2 {\left(\frac{8}{x{y}^4}\right)}^{-1}$

Use the Quotient to a Power Property, ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

$$\begin{gathered} {\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}={\left(4{)}^2{x}^2(y^3\right)}^2 \frac{8^{-1}}{x^{-1}{\left(y^4\right)}^{-1}} \\ {\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}=16x^2{y}^6 \frac{8^{-1}}{x^{-1}{y}^{-4}} \end{gathered}$$

Part (d) Step 2. Use the definition of negative exponent.

${\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}=16x^2{y}^6 ·\frac{x{y}^4}{8}$

Simplify.

${\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}=2x^2{y}^6 ·x{y}^4$

${\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}=2x^{2+1}{y}^{6+4}$

${\left(\frac{4x^3{y}^2}{x^2{y}^{-1}}\right)}^2 {\left(\frac{8x{y}^{-3}}{x^{2}y}\right)}^{-1}=2x^3{y}^{10}$