Q. 111 - Intermediate Algebra

Question

(a)$a^3.a^{-3}$

(b)$\left(u{v}^{-2}\right)\left(u^{-5}{v}^{-3}\right)$

(c)$(-4r^{-2}{s}^{-8})\left(9r^4{s}^3\right)$

Step-by-Step Solution

Answer

(a)$1$

(b)$u^{-4}{v}^{-5}$

(c)$-36r^2{s}^{-5}$

Part (a) Step 1 . Given Information

We are given the expression :

$a^3.a^{-3}$, we need to simplify the expression

Part (a) Step 2 . Product Property and Zero Exponent Property

If $a$ is a real number and $m$ and $n$ are integers, then :

$a^m.a^n=a^{m+n}$

       $a^0=1$

Part (a) Step 3 . Simplify the expression

Using the Product Property :

$a^3.a^{-3}=a^{3+(-3)}$

   $a^{3-3}=a^0$

        $a^0=1$ (zero exponent property)

Part (b) Step 1 . Given Information

We are given the expression :

$\left(u{v}^{-2}\right)\left(u^{-5}{v}^{-3}\right)$, we need to simplify the expression

Part (b) Step 2 . Product Property

If $a$ is a real number and $m$ and $n$ are integers, then :

$a^m.a^n=a^{m+n}$

Part (b) Step 3 . Simplify the expression

Using the Product Property :

$\left(u{v}^{-2}\right)\left(u^{-5}{v}^{-3}\right)=u^{1+(-5)}{v}^{-2+(-3)}$

       $u^{1-5}{v}^{-2-3}=u^{-4}{v}^{-5}$

Part (c) Step 1 . Given Information

We are given the expression :

$(-4r^{-2}{s}^{-8})\left(9r^4{s}^3\right)$, we need to simplify the expression

Part (c) Step 2 . Product Property

If $a$ is a real number and $m$ and $n$are integers, then :

$a^m.a^n=a^{m+n}$

Part (c) Step 3 . Simplify the expression

Using the Product Property :

$(-4r^{-2}{s}^{-8})\left(9r^4{s}^3\right)=(-4.9r^{-2+4})\left(s^{-8+3}\right)$

                                $=-36r^2{s}^{-5}$