Question

Simplify each expression:

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

Short 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-by-step solution

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.role="math" localid="1644763107906" $a^{-n}=\frac{1}{a^n}$

$z^{-4}·z^{-5}=\frac{1}{z^9}$

Part (b) Step 1. The given expression is (p6 q−2)(p−9q−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 (3u−5v7)(−4u4v−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.$a^{-n}=\frac{1}{a^n}$

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

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