Q. 5.33 - Intermediate Algebra

Question

Simplify each expression:

$ⓐ z^{-4}·z^{-5} ⓑ (p^6 q^{-2})\left(p^{-9}{q}^{-1}\right) ⓒ \left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})$

Step-by-Step Solution

Answer

The simplified expression of $z^{-4}·z^{-5}=\frac{1}{z^9}$

The simplified expression of $(p^6 q^{-2})\left(p^{-9}{q}^{-1}\right)=\frac{1}{p^3·q^3}$

The simplified expression of $\left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})=\frac{-12v^5}{u}$

Step 1. Given Information

In the given question we have to simplify each expression:

$ⓐ z^{-4}·z^{-5} ⓑ (p^6 q^{-2})\left(p^{-9}{q}^{-1}\right) ⓒ \left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})$

Part (a) Step 1. The given expression is z − 4 · z − 5

Add the exponents, since the bases are the same.

$z^{-4}·z^{-5}=z^{-4-5}$

Simplify.

$z^{-4}·z^{-5}=z^{-9}$

Use the definition of a negative exponent i.e.  $a^{-n}=\frac{1}{a^n}$

$z^{-4}·z^{-5}=\frac{1}{z^9}$

Part (b) Step 1. The given expression is ( p 6   q − 2 ) ( p − 9 q − 1 )

Use the Commutative Property to get like bases together.

$(p^6 q^{-2})\left(p^{-9}{q}^{-1}\right)=(p^6 p^{-9})·\left(q^{-2}{q}^{-1}\right)$

Add the exponents, since the bases are the same.

$(p^6 q^{-2})\left(p^{-9}{q}^{-1}\right)=p^{6-9}·q^{-2-1}$

Simplify.

$(p^6 q^{-2})\left(p^{-9}{q}^{-1}\right)=p^{-3}·q^{-3}$

Use the definition of a negative exponent, i.e. $a^{-n}=\frac{1}{a^n}$

$(p^6 q^{-2})\left(p^{-9}{q}^{-1}\right)=\frac{1}{p^3·q^3}$

Part (c) Step 1. The given expression is ( 3 u − 5 v 7 ) ( − 4 u 4 v − 2 )

Use the Commutative Property to get like bases together.

$\left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})=3·(-4)·\left(u^{-5}{u}^4\right)\left(v^7{v}^{-2}\right)$

Add the exponents, since the bases are the same.

$\left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})=3·(-4)·\left(u^{-5+4}\right)\left(v^{7-2}\right)$

Simplify.

$\left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})=-12·\left(u^{-1}\right)\left(v^5\right)$

Use the definition of a negative exponent i.e. $a^{-n}=\frac{1}{a^n}$

$\left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})=-12·\left(\frac{1}{u}\right)\left(v^5\right)$

$\left(3u^{-5}{v}^7\right)(-4u^4{v}^{-2})=\frac{-12v^5}{u}$