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Chapter 2: Problem 29

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

Short Answer

Expert verified
The simplified expression is \(x^{26} y^{\frac{52}{5}}\).

Step by step solution

01

Rewrite the expression with simplified exponents

Using the rule \((a^{m})^{n}= a^{mn}\), we can simplify the exponents in the given expression: \[ \frac{\left(x^{2} y^{4 / 5}\right)^{3}}{\left(x^{5} y^{2}\right)^{-4}} \] Becomes: \[ \frac{x^{2 \cdot 3} y^{(4 / 5) \cdot 3}}{x^{5 \cdot (-4)} y^{2 \cdot (-4)}} \]
02

Perform the multiplication in the exponents

Now, perform the multiplications in the exponents: \[ \frac{x^{6} y^{12 / 5}}{x^{-20} y^{-8}} \]
03

Combine the variables using exponent properties

Using the rule \(\frac{a^{m}}{a^{n}} = a^{m-n}\), we can combine the variables as follows: \[ x^{6-(-20)} y^{(12 / 5)-(-8)} \]
04

Perform the subtraction in the exponents

Now, perform the subtractions in the exponents: \[ x^{6+20} y^{(12 / 5)+8} \]
05

Simplify the exponents

Simplify the exponents, making sure to add the whole numbers and fractions separately: \[ x^{26} y^{(12 / 5)+40 / 5} \] Which becomes: \[ x^{26} y^{52 / 5} \]
06

Final simplified expression

The simplified expression is: \[ x^{26} y^{52 / 5} \]

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Most popular questions from this chapter

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

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 \(s ?\)

Explain why the composition of two polynomials is a polynomial.

Find a polynomial \(p\) of degree 3 such that \(-1,2,\) and 3 are zeros of \(p\) and \(p(0)=1\).

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?]
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