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Chapter 2: Problem 17

Simplify the given expression by writing it as a power of a single variable. $$ y^{4}\left(y^{2}\left(y^{5}\right)^{2}\right)^{3 / 5} $$

Short Answer

Expert verified
The simplified expression is \(y^{\frac{56}{5}}\).

Step by step solution

01

Identify the exponent properties

We have the following properties of exponents that will help us simplify the expression: 1. Power of a power: \((x^a)^b = x^{a \cdot b}\) 2. Product of powers: \(x^a \cdot x^b = x^{a+b}\)
02

Apply the power of a power property

First, we will focus on the inner part of the expression, \((y^5)^2\). Using the power of a power property, we can simplify this part as follows: \((y^5)^2 = y^{5 \cdot 2} = y^{10}\) Now, our expression becomes: \(y^4\left(y^2\left(y^{10}\right)\right)^{3 / 5}\)
03

Apply the product of powers property

Next, we will simplify the expression inside the parentheses. We have \(y^2\) multiplied by \(y^{10}\), so using the product of powers property, we can simplify this as follows: \(y^2 \cdot y^{10} = y^{2+10} = y^{12}\) Now, our expression becomes: \(y^4\left(y^{12}\right)^{3 / 5}\)
04

Apply the power of a power property again

Now, we will focus on the expression \((y^{12})^{3/5}\). Using the power of a power property, we can simplify this part as follows: \((y^{12})^{3/5} = y^{12 \cdot 3/5} = y^{36/5}\) Now, our expression becomes: \(y^4 \cdot y^{36/5}\)
05

Apply the product of powers property again

Finally, we have \(y^4\) multiplied by \(y^{36/5}\), so using the product of powers property, we can simplify the expression to its final form as follows: \(y^4 \cdot y^{36/5} = y^{4 + 36/5} = y^{(\frac{20}{5}) + (\frac{36}{5})} = y^{\frac{56}{5}}\)
06

Final Simplified Expression

The given expression simplifies to the power of a single variable as: \(y^{\frac{56}{5}}\).

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Most popular questions from this chapter

Suppose \(p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n},\) where \(a_{0}, a_{1}, \ldots, a_{n}\) are integers. Suppose \(m\) is a nonzero integer that is a zero of \(p\). Show that \(a_{0} / m\) is an integer. [This result shows that to find integer zeros of a polynomial with integer coefficients, we need only look at divisors of its constant term.]

Find a polynomial \(p\) of degree 3 such that \(-2,-1,\) and 4 are zeros of \(p\) and \(p(1)=2\).

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

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{6 x^{4}+4 x^{3}-7}{2 x^{4}+3 x^{2}+5} $$

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{6}-4 x^{2}+5}{x^{2}-3 x+1} $$
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