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Chapter 10: Problem 368

In the following exercises, simplify. $$\left(y^{-5}\right)^{4}$$

Short Answer

Expert verified
The simplified form is \(\frac{1}{y^{20}}\).

Step by step solution

01

Identify the Law of Exponents

When simplifying expressions with exponents, use the law of exponents that states \((a^m)^n = a^{m \times n}\). This means we will multiply the exponents.
02

Apply the Law of Exponents

Apply the law of exponents to the expression \[ (y^{-5})^4 \]. According to the law, multiply the exponents -5 and 4.
03

Calculate the Multiplication

Calculate \(-5 \times 4\), which gives \(-20\). Therefore, the expression becomes \(y^{-20}\).
04

Simplify the Expression

The simplified form of \(y^{-20}\) is \(\frac{1}{y^{20}}\) because any variable with a negative exponent can be written as the reciprocal with a positive exponent.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a common task in algebra. It involves reducing a complex expression to its simplest form. This makes it easier to understand and work with. When you simplify, you aim to eliminate any unnecessary terms or operations.
For example, let's consider the expression: \(\left(y^{-5}\right)^{4}\). We want to apply the laws of exponents to simplify it.
Remember to always follow the proper order of operations and apply the correct exponent rules. This ensures a smooth and error-free simplification.
Negative Exponents
A negative exponent means that the base will be on the denominator side of a fraction. It doesn't mean the whole expression becomes negative. Instead, it results in a reciprocal.
For example, if we have \(y^{-5}\), this can be simplified to \(\frac{1}{y^{5}}\). So, a negative exponent in the numerator sends the base to the denominator with a positive exponent.
In our exercise, \(y^{-20}\) is simplified to \(\frac{1}{y^{20}}\). This follows the rule of negative exponents, making it clearer and simpler.
Multiplication of Exponents
Multiplication of exponents comes into play when you have a power raised to another power. The law of exponents tells us that \((a^m)^n = a^{m \times n}\). You multiply the exponents together.
For the expression \(\left(y^{-5}\right)^4\), we multiply -5 by 4. This results in -20.
Hence, the expression transforms to \(y^{-20}\). Once combined with the rules of negative exponents, we get the simplified form \(\frac{1}{y^{20}}\).
This process highlights how multiplication of exponents can greatly simplify an expression.

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