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Chapter 1: Problem 40

Convert to radical notation. \(\left(y^{2}+7\right)^{-1 / 4}\)

Short Answer

Expert verified
Convert \((y^{2}+7)^{-1/4}\) to \(\frac{1}{\root{4}\relax{(y^{2}+7)}}\).

Step by step solution

01

- Identify the Exponent

The expression given is \((y^{2}+7)^{-1/4}\). The exponent here is \(-1/4\).
02

- Apply Negative Exponent Rule

A negative exponent indicates the reciprocal of the base. So \((y^{2}+7)^{-1/4}\) becomes \(\frac{1}{(y^{2}+7)^{1/4}}\).
03

- Convert Exponent to Radical Form

An exponent of \(1/4\) is the same as the fourth root. Thus, \((y^{2}+7)^{1/4}\) can be written as \(\root{4}\relax{(y^{2}+7)}\).
04

- Combine All Components

Putting it all together, the expression \(\frac{1}{(y^{2}+7)^{1/4}}\) converts to \(\frac{1}{\root{4}\relax{(y^{2}+7)}}\).

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Key Concepts

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

Negative Exponent Rule
Understanding the negative exponent rule is crucial for manipulating and simplifying expressions in algebra. When an expression has a negative exponent, it indicates that the base should be taken as a reciprocal, or in other words, inverted. For instance, \((x^{-a} = \frac{1}{x^{a}} )\). This transformation allows us to work with positive exponents, which are easier to handle and interpret. Hence, in the original problem, \((y^{2}+7)^{-1/4} \equiv \frac{1}{(y^{2}+7)^{1/4}}\). This step makes the next computations more straightforward.
Radical Form
Converting expressions to their radical form helps in visualizing the root operations. When dealing with fractional exponents, remember that they can be represented as radicals. Specifically, if we have an exponent like \(1/4\), it means taking the fourth root. For example, \(x^{1/4}\) is equivalent to \(\root{4}\relax{x}\). So, applying this to our expression, \((y^{2}+7)^{1/4}\) becomes \(\root{4}\relax{(y^{2}+7)}\). This transformation from exponential to radical form is essential in many areas of math, including solving equations and simplifying expressions.
Reciprocal
The concept of a reciprocal is closely tied to the negative exponent rule. A reciprocal flips a number or an expression upside down. For example, the reciprocal of \(a\) is \(\frac{1}{a}\). This idea is applied directly when dealing with negative exponents. Thus, in our given problem, the reciprocal transformation changes \(\frac{1}{(y^{2}+7)^{1/4}}\) into its radical form. Using reciprocal relationships simplifies complex expressions by making operations more manageable and interpretations more intuitive.

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