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Chapter 2: Problem 27

Simplify the given expression. $$ \frac{\left(x^{-2}\right)^{3} y^{8}}{x^{-5}\left(y^{4}\right)^{-3}} $$

Short Answer

Expert verified
The simplified expression is \( \frac{y^{20}}{x} \).

Step by step solution

01

1. Rewrite the expression using the power of a power rule

: First, let's simplify the expression using the power of a power rule. This rule states that for any nonzero number 'a' and integers m and n, \((a^m)^n = a^{mn}\). Using this rule, the given expression can be rewritten as: \[ \frac{(x^{-2(3)}) y^8}{x^{-5} (y^{4(-3)})} \]
02

2. Simplify the exponents:

: Now, let's multiply the exponents to get: \[ \frac{x^{-6} y^8}{x^{-5} y^{-12}} \]
03

3. Apply the rule of exponents in division:

: Next, we will apply the rule of exponents in division, which states that for any nonzero number 'a' and integers m and n, \(\frac{a^m}{a^n} = a^{m-n}\). Applying this rule to both x and y, we get: \[ x^{-6 - (-5)} y^{8 - (-12)} \]
04

4. Simplify the expression:

: Finally, simplify the expression by performing the subtraction in the exponents: \[ x^{-1} y^{20} \]
05

5. Rewrite the final expression using positive exponents

: Since the expression has a negative exponent, rewrite the expression using the definition of negative exponents, \(a^{-m} = \frac{1}{a^m}\), to obtain the final simplified expression: \[ \frac{y^{20}}{x} \]

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Most popular questions from this chapter

Let \(p\) be the polynomial defined by $$p(x)=x^{6}-87 x^{4}-92 x+2$$. (a)Use a computer or calculator to sketch a graph of \(p\) on the interval [-5,5] . (b) Is \(p(x)\) positive or negative for \(x\) near \(\infty ?\) (c) Is \(p(x)\) positive or negative for \(x\) near \(-\infty ?\) (d) Explain why the graph from part (a) does not accurately show the behavior of \(p(x)\) for large values of \(x\).

Suppose \(M\) and \(N\) are odd integers. Explain why $$ x^{2}+M x+N $$ has no rational zeros.

Explain why the polynomial \(p\) defined by $$ p(x)=x^{6}+100 x^{2}+5 $$ has no real zeros.

Without doing any calculations or using a calculator, explain why $$ x^{2}+87559743 x-787727821 $$ has no integer zeros. [Hint: If \(x\) is an odd integer, is the expression above even or odd? If \(x\) is an even integer, is the expression above even or odd?]

A textbook states that the rabbit population on a small island is observed to be $$ 1000+120 t-0.4 t^{4} $$ where \(t\) is the time in months since observations of the island began. Explain why the formula above cannot correctly give the number of rabbits on the island for large values of \(t\).
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