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Chapter 10: Problem 97

Simplify each exponential expression. $$\frac{7 x^{-1} y}{14\left(x^{5} y^{2}\right)^{-2}}$$

Short Answer

Expert verified
The simplified expression is \(\frac{1}{2} x^9 y^5\).

Step by step solution

01

Simplify the Exponent in the Denominator

First, recognize that the expression in the denominator has a negative exponent: \((x^{5} y^{2})^{-2}\). Applying the rule for negative exponents, we have: \((x^{5} y^{2})^{-2} = \frac{1}{(x^{5} y^{2})^{2}}\). This means the denominator becomes \(14 / (x^{10} y^{4})\), because you square each part.
02

Rewrite the Expression with Positive Exponents

Given the expression \(\frac{7x^{-1}y}{14 / (x^{10} y^{4})}\), we can multiply by the reciprocal to get rid of the fraction in the denominator, so: \(\frac{7x^{-1}y}{14} \times (x^{10} y^{4})\). This simplifies to: \(\frac{7x^{-1+10} y^{1+4}}{14}\).
03

Simplify the Fractions and Exponents

Simplify the coefficient \(\frac{7}{14}\) to \(\frac{1}{2}\), then apply the laws of exponents: \(x^{-1+10} = x^9\) and \(y^{1+4} = y^5\). So the expression now is: \(\frac{1}{2}x^9 y^5\).
04

Write the Simplified Expression

Therefore, the simplified expression is \(\frac{1}{2} x^9 y^5\).

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Key Concepts

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

Understanding Negative Exponents
A negative exponent might seem daunting at first, but it's actually a simple concept! When you see a number or variable raised to a negative power, it just means you take the reciprocal. For instance, if you have an expression like \(x^{-1}\), instead of just \(x\), think of it as \(\frac{1}{x}\).
This transformation applies to any negative exponent:
  • \(a^{-n} = \frac{1}{a^n}\)
  • \(x^{-3} = \frac{1}{x^3}\)
Remember, negative exponents help simplify expressions by shifting the base from the numerator to the denominator or vice versa.
In our exercise, \((x^5y^2)^{-2}\) changes to \(\frac{1}{(x^{10}y^4)}\). This is because squaring both \(x^5\) and \(y^2\) gives us \(x^{10}\) and \(y^{4}\), respectively.
Mastering the Laws of Exponents
Understanding the laws of exponents is crucial when simplifying any exponent expression. There are a few basic laws that will make your life easier when dealing with exponents:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m\cdot n}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
These rules help combine, multiply, and divide expressions with exponents efficiently!
In the exercise, we applied the product of powers: \(x^{-1} \cdot x^{10} = x^{9}\) and \(y^{1} \cdot y^{4} = y^{5}\). These steps show how we can handle exponents, making them user-friendly and simplifying them step by step.
Fractions with Exponents Explained
Fractions involving exponents can look tricky, but they follow the same basic rules.
When simplifying fractions with exponents, the idea is to express everything in terms of positive exponents. Use these guiding tips:
  • Reciprocals convert negative exponents to positive.
  • To eliminate a fraction in the denominator, multiply by its reciprocal.
In our exercise, we initially had the fraction \(\frac{7x^{-1}y}{14/(x^{10}y^4)}\). By multiplying by the reciprocal \((x^{10}y^4)\), the fraction disappeared, making it easier to use the laws of exponents.
Spearheading through such expressions with these basic rules ensures an easier and efficient way to solve and simplify any fractional exponent problems.

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Most popular questions from this chapter

Multiply. $$ x^{1 / 2}\left(x^{1 / 2}+x^{3 / 2}\right) $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y^{2}} \cdot \sqrt[6]{y} $$

Use rational expressions to write as a single radical expression. $$ \sqrt[4]{5} \cdot \sqrt[3]{x} $$

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as \(8 \cdot 3,\) because 8 is a perfect cube. $$ 80 $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(a^{-2} b^{3}\right)^{1 / 8}}{\left(a^{-3} b\right)^{-1 / 4}} $$
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