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Chapter 10: Problem 43

Simplify the expression. $$\left(\frac{10}{y}\right)^{2}$$

Short Answer

Expert verified
\(\frac{100}{y^2}\)

Step by step solution

01

Identify the Base and Exponent

The expression can be identified with a base of \(\frac{10}{y}\) and an exponent of 2.
02

Apply the Power of a Quotient Rule

Using the rule \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), the expression becomes \(\frac{10^2}{y^2}\).
03

Simplify Exponents

Calculate the exponent on the numerator: \(10^2 = 100\). Thus, the expression simplifies to \(\frac{100}{y^2}\).

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Key Concepts

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

Simplifying Algebraic Expressions
Simplifying algebraic expressions is a fundamental skill in algebra. It involves breaking down a complex expression into a simpler form. This process can include operations like combining like terms, using distributive properties, and applying exponent rules. When done correctly, simplification helps in solving equations more easily and understanding the relationships between variables. For example, in the expression \((\frac{10}{y})^{2}\), simplifying helps in clearly understanding how the numerator and denominator interact when raised to a power.
Power of a Quotient Rule
The power of a quotient rule is crucial for simplifying expressions involving fractions. It states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). This means you raise both the numerator and the denominator to the power of the exponent separately. Using our example, \((\frac{10}{y})^2\), the rule implies: \((\frac{10}{y})^2 = \frac{10^2}{y^2}\). This step simplifies the calculation and makes it easier to manage larger expressions.
Understanding Exponents
Exponents are a shorthand way of indicating that a number should be multiplied by itself a certain number of times. In the context of \(\frac{10^2}{y^2}\), exponent 2 means 10 is multiplied by 10, resulting in 100. Similarly, y is multiplied by y, yielding \(y^2\). Remembering this can help when simplifying fractions with exponents. Here, the expression becomes \((\frac{10}{y})^2 = \frac{100}{y^2}\).

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Most popular questions from this chapter

Which is greater? \(\left(2^{3}\right)^{2}\) or \(\left(2^{2}\right)^{3}\)

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