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Chapter 2: Problem 26

Simplify the given expression. $$ \frac{x^{11}\left(y^{3}\right)^{2}}{\left(x^{3}\right)^{5}\left(y^{2}\right)^{4}} $$

Short Answer

Expert verified
The simplified expression is \(\frac{1}{x^4 y^2}\).

Step by step solution

01

Simplify the numerator.

To simplify the numerator, apply the Power Rule to the expression \((y^3)^2\). This is equivalent to raising \(y^3\) to the power of 2. \[(y^3)^2 = y^{(3)(2)} = y^6\] So, the simplified numerator becomes: \[x^{11}y^6\]
02

Simplify the denominator.

To simplify the denominator, apply the Power Rule to both exponential expressions \(\left(x^3\right)^5\) and \(\left(y^2\right)^4\). This means we will raise \(x^3\) to the power of 5 and \(y^2\) to the power of 4. \[\left(x^3\right)^5 = x^{(3)(5)} = x^{15}\] \[\left(y^2\right)^4 = y^{(2)(4)} = y^{8}\] So, the simplified denominator becomes: \[x^{15}y^8\]
03

Divide the simplified numerator by the simplified denominator.

Now, we will divide the simplified numerator \(x^{11}y^6\) by the simplified denominator \(x^{15}y^8\). This involves applying the Division Rule to both \(x\) and \(y\), which means we will subtract the exponents of \(x\) and \(y\) in the denominator from their respective exponents in the numerator. \[\frac{x^{11}y^6}{x^{15}y^8} = x^{11-15}y^{6-8} = x^{-4}y^{-2}\]
04

Express the simplified expression with positive exponents.

The final simplified expression has negative exponents. To write the expression with positive exponents, we will flip the \(x\) and \(y\) terms with negative exponents from the numerator to the denominator (or vice versa) and change the signs of their exponents. \[x^{-4}y^{-2} = \frac{1}{x^4 y^2}\] The simplified expression is: \[\frac{1}{x^4 y^2}\]

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Most popular questions from this chapter

Suppose \(r\) is the function with domain \((0, \infty)\) defined by $$ r(x)=\frac{1}{x^{4}+2 x^{3}+3 x^{2}} $$ for each positive number \(x\). (a) Find two distinct points on the graph of \(r\). (b) Explain why \(r\) is a decreasing function on \((0, \infty)\). (c) Find two distinct points on the graph of \(r^{-1}\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r-s)(x) $$

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{2 x+1}{x-3} $$

Factor \(x^{16}-y^{8}\) as nicely as possible.

Write the domain of the given function \(r\) as a union of intervals. $$ r(x)=\frac{4 x^{7}+8 x^{2}-1}{x^{2}-2 x-6} $$
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