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Chapter 6: Problem 90

Evaluate the following expressions: (a) \(\left(-12 x^{10} y^{9} z^{5}\right) /\left(3 x^{2} y^{3} z^{6}\right)\) (b) \(\left(-16 x^{16} y^{6} z^{4}\right) /\left(-4 x^{4} y^{2} z^{7}\right)\)

Short Answer

Expert verified
In summary, the evaluated expressions are: (a) \(-4 x^8 y^6 z^{-1}\) (b) \(4 x^{12} y^4 z^{-3}\)

Step by step solution

01

(a) Evaluating the coefficients

For expression (a), we start by dividing the coefficients: \[\frac{-12}{3} = -4\]
02

(a) Dividing the variables

Now we divide the variables one by one using the properties of exponents: \[\frac{x^{10}}{x^{2}} = x^{10-2} = x^8\] \[\frac{y^{9}}{y^{3}} = y^{9-3} = y^6\] \[\frac{z^{5}}{z^{6}} = z^{5-6} = z^{-1}\] Now, let's combine the coefficients and variables to get the result for expression (a).
03

(a) Final result for expression (a)

The final result for expression (a) is: \[ \frac{-12 x^{10} y^{9} z^{5}}{3 x^{2} y^{3} z^{6}} = -4 x^8 y^6 z^{-1} \] Now, let's evaluate expression (b).
04

(b) Evaluating the coefficients

For expression (b), we start by dividing the coefficients: \[\frac{-16}{-4} = 4\]
05

(b) Dividing the variables

Now we divide the variables one by one using the properties of exponents: \[\frac{x^{16}}{x^{4}} = x^{16-4} = x^{12}\] \[\frac{y^{6}}{y^{2}} = y^{6-2} = y^4\] \[\frac{z^{4}}{z^{7}} = z^{4-7} = z^{-3}\] Now, let's combine the coefficients and variables to get the result for expression (b).
06

(b) Final result for expression (b)

The final result for expression (b) is: \[ \frac{-16 x^{16} y^{6} z^{4}}{-4 x^{4} y^{2} z^{7}} = 4 x^{12} y^4 z^{-3} \] In summary, the evaluated expressions are: (a) \(-4 x^8 y^6 z^{-1}\) (b) \(4 x^{12} y^4 z^{-3}\)

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Most popular questions from this chapter

Simplify the quotient \(\left(x^{-2} y^{4}\right)^{3} /(x y)^{-3}\).

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