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Chapter 9: Problem 93

For Exercises, simplify. $$\frac{\left(x^{-2} y\right)^{2}}{x^{2} y^{3}}$$

Short Answer

Expert verified
The simplified form of the fraction is \( \frac{1}{x^{2}y} \).

Step by step solution

01

Convert Negative Exponents

Begin by converting the negative exponent to a positive exponent. Recall that \(x^{-n} = 1/x^{n}\). Therefore, \(x^{-2} = 1/x^{2}\) and the fraction becomes: \(\frac{\left(1/x^{2} y\right)^{2}}{x^{2} y^{3}}\).
02

Compute Exponent Power

Now, solve the exponent of the numerator. Applying the power to each part inside the parentheses gives \(1^2/x^{4} y^{2}=\frac{1}{x^{4}}y^{2}\). So our fraction is now \(\frac{\frac{1}{x^{4}}y^{2}}{x^{2} y^{3}}\).
03

Simplify Fraction

In this step, we will simplify the multiplication in the numerator and denominator. For the same base, the rule is: \(x^{a}/x^{b}=x^{a-b}\). As such, we simplify \(x^{4}/x^{2}=x^{2}\) and \(y^{2}/y^{3}=1/y\). We then end up with \(1/x^{2}*1/y=\frac{1}{x^{2}y}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Negative Exponents
Negative exponents might seem tricky at first, but they are just a way to express reciprocals of numbers or variables raised to positive exponents.
Understanding the basics can simplify your journey through algebra.
  • When you encounter an expression like \( x^{-n} \), it signifies a reciprocal, which can be rewritten as \( \frac{1}{x^{n}} \).
  • This conversion is crucial, especially when simplifying complex expressions where negative exponents appear. Remember that negative exponents change the position of the base in terms of being a numerator or denominator.
  • In the original expression \( \frac{(x^{-2} y)^{2}}{x^{2} y^{3}} \), converting \( x^{-2} \) to \( \frac{1}{x^{2}} \) helps in transforming the entire problem to a more solvable format.
Grasping this concept will assist you in progressing to problems involving powers and roots, as negative exponents connect directly with these areas.
Steps to Fraction Simplification
Simplifying fractions is a fundamental skill in algebra that involves reducing the fraction to its simplest form with the intention of making it easier to understand or compute.
  • Start by simplifying terms with similar bases; these allow the use of laws of exponents like \( \frac{x^{a}}{x^{b}} = x^{a-b} \).
  • In the given exercise, for example, simplifying \( \frac{x^{4}}{x^{2}} \) resulted in \( x^{2} \), thanks to exponent subtraction.
  • The same applies to \( y \) terms, where \( \frac{y^{2}}{y^{3}} = \frac{1}{y} \).
This step-by-step technique not only helps in algebra but also builds accuracy in solving everyday mathematical problems.
Applying the Power of a Quotient Property
Understanding the power of a quotient property is vital for simplifying expressions like the one in our exercise.
This property deals with raising both the numerator and the denominator of a fraction to a power, which enables the simplification of more complex terms.
  • The rule states that \( \left(\frac{a}{b}\right)^{n} = \frac{a^n}{b^n} \), which allows even distribution of the power to each part of the fraction.
  • For example, in our task, \( \left(x^{-2} y\right)^{2} \) resulted in \( \frac{1}{x^{4}} y^{2} \) by applying this rule to both \( x \) and \( y \) terms.
  • This method ensures that each component of the fraction is correctly simplified following the power rule.
By mastering the power of quotient, you can tackle more complex fractions with ease, providing a strong foundation for advanced algebraic manipulations.

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Most popular questions from this chapter

Multiply. $$(2 x-9 y)(8 x-3 y)$$

If \(x\) is a nonzero real number, is \(x^{-3}\) always positive, always negative, or positive or negative depending on whether \(x\) is positive or negative? Explain your answer.

If \(n\) is a positive integer greater than \(1,\) how many zeros appear before the decimal point when \(1.35 \times 10^{n}\) is written in decimal notation?

Solve. $$(-4.8)^{x}=1$$

If \(x\) is a nonzero real number, is \(x^{-2}\) always positive, always negative, or positive or negative depending on whether \(x\) is positive or negative? Explain your answer.
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