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Chapter 10: Problem 249

In the following exercises, simplify. $$\left(\frac{x}{y}\right)^{10}$$

Short Answer

Expert verified
\frac{x^{10}}{y^{10}}

Step by step solution

01

Identify the Expression

Consider the given expression \(\frac{x}{y}\right)^{10}\).
02

Apply the Power Rule

Use the power rule for exponents, which states that \(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). Apply this rule to the expression \(\frac{x}{y}\right)^{10}\).
03

Simplify the Expression

Following the power rule, simplify the expression to \(\frac{x^{10}}{y^{10}}\right)\).
04

Final Simplified Form

The expression \(\frac{x^{10}}{y^{10}}\right)\) is the simplified form.

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Key Concepts

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

Power Rule for Exponents
The power rule for exponents is a fundamental concept in algebra. It helps you simplify expressions where an entire fraction is raised to an exponent. To apply this rule, follow these steps: Take each component of the fraction separately. Raise both the numerator and the denominator to the given power. This means if you have \(\frac{a}{b}\)^n, it becomes \(\frac{a^n}{b^n}\). Simplifying like this makes complex expressions much easier to handle. For example, in the exercise, \(\frac{x}{y}\)^{10} becomes \(\frac{x^{10}}{y^{10}}\).
Simplification
Simplification in algebra means reducing an expression to its simplest form. This makes it easier to understand and solve. When simplifying, you should always follow the basic rules of algebra. For example, combine like terms and use the power rules for exponents and fractions. Always look out for common factors you can cancel out.
In the exercise, after applying the power rule, we simplified \(\frac{x}{y}\)^{10} to \(\frac{x^{10}}{y^{10}}\). This form is simpler because it follows all algebraic rules, making it easy to work with.
Fractions in Algebra
Fractions are a crucial part of algebra. They allow you to represent parts of a whole and perform operations between those parts. Simplifying fractions involves a few key steps:
  • First, break down the numerator and the denominator.
  • Second, look for common factors.
  • Lastly, cancel out those common factors to simplify the fraction.

When dealing with exponents, like in the exercise \(\frac{x}{y}\)^{10}, breaking down and applying the power rule simplifies complex fraction operations. In this case, the fraction \(\frac{x}{y}\)^ becomes \(\frac{x^{10}}{y^{10}}\) once you apply the power rule correctly. This shows the power rule's utility in making expressions simpler and easier to understand.

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Most popular questions from this chapter

Simplify each expression. $$n \cdot n^{4}$$

Simplify. $$\frac{\left(n^{5}\right)^{3}}{\left(n^{2}\right)^{8}}$$

Multiply using any method. $$(4 y-1)\left(6 y^{2}-12 y+5\right)$$

Find the greatest common factor. $$9 y^{4}, 21 y^{5}, 15 y^{6}$$

Simplify. $$(8 n)^{-1}$$
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