Q. 5.34 - Intermediate Algebra

Question

Simplify each expression:

$ⓐ c^{-8}·c^{-7} ⓑ (r^5 s^{-3})\left(r^{-7}{s}^{-5}\right) ⓒ (-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})$

Step-by-Step Solution

Answer

The simplified expression of $c^{-8}·c^{-7}=\frac{1}{c^{15}}$

The simplified expression of $(r^5 s^{-3})\left(r^{-7}{s}^{-5}\right)=\frac{1}{r^2{s}^8}$

The simplified expression of $(-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})=\frac{30d^3}{c^8}$

Step 1. Given Information

In the given question we have to simplify each expression:

$ⓐ c^{-8}·c^{-7} ⓑ (r^5 s^{-3})\left(r^{-7}{s}^{-5}\right) ⓒ (-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})$

Part (a) Step 1. The given expression is c − 8 · c − 7

Add the exponents, since the bases are the same.

$c^{-8}·c^{-7}=c^{-8-7}$

Simplify.

$c^{-8}·c^{-7}=c^{-15}$

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

$c^{-8}·c^{-7}=\frac{1}{c^{15}}$

Part (b) Step 1. The given expression is ( r 5   s − 3 ) ( r − 7 s − 5 )

Use the Commutative Property to get like bases together.

$(r^5 s^{-3})\left(r^{-7}{s}^{-5}\right)=(r^5 r^{-7})\left(s^{-3}{s}^{-5}\right)$

Add the exponents, since the bases are the same.

$(r^5 s^{-3})\left(r^{-7}{s}^{-5}\right)=\left(r^{5-7}\right)\left(s^{-3-5}\right)$

Simplify.

$(r^5 s^{-3})\left(r^{-7}{s}^{-5}\right)=\left(r^{-2}\right)\left(s^{-8}\right)$

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

$(r^5 s^{-3})\left(r^{-7}{s}^{-5}\right)=\left(\frac{1}{r^2}\right)\left(\frac{1}{s^8}\right)$

$(r^5 s^{-3})\left(r^{-7}{s}^{-5}\right)=\frac{1}{r^2{s}^8}$

Part (c) Step 1. The given expression is ( - 6 c − 6 d 4 ) ( − 5 c - 2 d − 1 )

Use the Commutative Property to get like bases together.

$(-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})= (-6)·(-5)\left(c^{-6}{c}^{-2}\right)\left(d^{-1}{d}^4\right)$

Add the exponents, since the bases are the same.

$(-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})= (-6)·(-5)\left(c^{-6-2}\right)\left(d^{-1+4}\right)$

Simplify.

$(-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})= 30\left(c^{-8}\right)\left(d^3\right)$

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

$(-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})= 30\left(\frac{1}{c^8}\right)\left(d^3\right)$

$(-6c^{-6}{d}^4)(-5c^{-2}{d}^{-1})=\frac{30d^3}{c^8}$