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Chapter 2: Problem 25

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

Short Answer

Expert verified
\( \frac{x}{y^4} \)

Step by step solution

01

Use the Power of a Power Rule

Since we have the expression \((x^2)^3\), we need to use the Power of a Power Rule, which states that \((a^m)^n = a^{mn}\). Thus, in our case, we have: \((x^2)^3 = x^{(2)(3)} = x^6\). The expression now becomes: $$ \frac{x^6 y^8}{x^5(y^4)^3} $$
02

Apply the Power of a Power Rule to \(y\)

Now let's apply the Power of a Power Rule to the term \((y^4)^3\), which gives us: \((y^4)^3 = y^{(4)(3)} = y^{12}\). Now the expression looks like: $$ \frac{x^6 y^8}{x^5y^{12}} $$
03

Use the Quotient of Powers Rule for \(x\)

The Quotient of Powers Rule states that \(\frac{a^m}{a^n} = a^{m-n}\). Let's apply this rule to the terms involving \(x\). We get: $$ \frac{x^6}{x^5} = x^{(6-5)} = x^1 $$ The expression now becomes: $$ \frac{x^1 y^8}{y^{12}} $$
04

Use the Quotient of Powers Rule for \(y\)

Finally, let's apply the Quotient of Powers Rule to the terms involving \(y\): $$ \frac{y^8}{y^{12}} = y^{(8-12)} = y^{-4} $$
05

Combine the results

Now we can combine our results and write the simplified expression: $$ x^1 y^{-4} $$ which is equivalent to: $$ \frac{x}{y^4} $$

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Most popular questions from this chapter

Suppose $$r(x)=\frac{x+1}{x^{2}+3} \quad \text { and } \quad s(x)=\frac{x+2}{x^{2}+5}$$ Find two distinct numbers \(x\) such that \(s(x)=\frac{1}{8}\).

Suppose \(q(x)=2 x^{3}-3 x+1\) (a) Show that the point (2,11) is on the graph of \(q\). (b) Show that the slope of a line containing (2,11) and a point on the graph of \(q\) very close to (2,11) is approximately 21 . [Hint: Use the result of Exercise \(17 .]\)

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 \(r^{-1}\).

Show that $$ (a+b)^{3}=a^{3}+b^{3} $$ if and only if \(a=0\) or \(b=0\) or \(a=-b\).

Suppose $$r(x)=\frac{x+1}{x^{2}+3} \quad \text { and } \quad s(x)=\frac{x+2}{x^{2}+5}$$ What is the domain of \(s ?\)
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