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Chapter 0: Problem 104

Simplify the expressions. $$ \frac{x^{-1 / 2} y}{x^{2} y^{3 / 2}} $$

Short Answer

Expert verified
The short answer is: \(\frac{1}{x^{5/2} \cdot y^{1/2}}\).

Step by step solution

01

Recall the properties of exponents

When dealing with exponential expressions, there are a few important exponent properties to remember: 1. \(a^{m} \cdot a^{n} = a^{m+n}\) 2. \(\frac{a^{m}}{a^{n}} = a^{m-n}\) 3. \(a^{-n} = \frac{1}{a^{n}}\)
02

Apply exponent properties and combine terms with the same base

Rewrite the expression using property 3 (Rewrite negative exponents as fractions) and then apply property 2 (Divide the exponent terms): \(\frac{x^{-1/2} y}{x^{2} y^{3 / 2}} = \frac{\frac{1}{x^{1/2}} \cdot y}{x^{2} \cdot y^{3 / 2}}\) Next, apply property 2 to x and y terms separately: \(= \frac{1}{x^{1/2}}\cdot \frac{y}{y^{3/2}} \cdot \frac{1}{x^2}\)
03

Simplify the expression using properties of exponents

Now, apply the properties of exponents on both \(x\) and \(y\) terms: For \(x\): \(\frac{1}{x^{1/2}} \cdot \frac{1}{x^{2}} = x^{(-1/2) -2}\) For \(y\): \(\frac{y}{y^{3/2}} = y^{1- (3/2)}\)
04

Combine the simplified terms of x and y

Now combine the simplified terms of \(x\) and \(y\) to find the final expression: \(x^{(-1/2) -2} \cdot y^{1- (3/2)} = x^{-5/2} \cdot y^{-1/2}\)
05

Write the final simplified expression

Finally, rewrite the expression using property 3 (Rewrite negative exponents as fractions): \[x^{-5/2} \cdot y^{-1/2} = \frac{1}{x^{5/2} \cdot y^{1/2}}\] The simplified expression is: \(\frac{1}{x^{5/2} \cdot y^{1/2}}\)

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Most popular questions from this chapter

Calculate each expression in Exencises giving the answer as \(a\) whole mumber or a fraction in lowest terms.\(3^{\wedge} 2+2^{\wedge} 2+1\)

Giving the answer as \(a\) whole number or a fraction in lowest terms. $$ 3+([4-2] \cdot 9) $$

Solve the equation for \(x\) (mentally, if possible). \(x-1=c x+d \quad(c \neq 1)\)

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(3\left(1+\frac{4}{100}\right)^{-3}\)

\(10 x\left(x^{2}+1\right)^{4}\left(x^{3}+1\right)^{5}+15 x^{2}\left(x^{2}+1\right)^{5}\left(x^{3}+1\right)^{4}\)
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