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Chapter 1: Problem 27

Convert to expressions with rational exponents. \(\frac{1}{\sqrt{x^{2}+7}}\)

Short Answer

Expert verified
(x^2 + 7)^{-\frac{1}{2}}

Step by step solution

01

Identify the Radical Expression

The given expression is a fraction with a radical in the denominator: \(\frac{1}{\text{something}}\). The 'something' part is \(\sqrt{x^2 + 7}\).
02

Rewrite the Radical as an Exponent

Recall that the square root \(\sqrt{a}\) can be rewritten as an exponent: \(a^{\frac{1}{2}}\). Therefore, \(\sqrt{x^2 + 7}\) can be rewritten as \((x^2 + 7)^{\frac{1}{2}}\).
03

Rewrite the Entire Expression

Using the rewritten exponent, the entire expression becomes \(\frac{1}{(x^2 + 7)^{\frac{1}{2}}}\).
04

Simplify the Expression

Recall that \(\frac{1}{a^n} = a^{-n}\). Apply this to the expression: \((x^2 + 7)^{-\frac{1}{2}}\).

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Key Concepts

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

Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, etc. In the exercise, we started with the radical expression \(\frac{1}{\text{something}}\), where 'something' was \(\sqrt{x^2 + 7}\).

To understand radical expressions better:
  • A square root, like \(\text{\sqrt{a}}\), means finding a number which, when squared, will give you \(\text{a}\).
  • A cube root, \(\text{\sqrt[3]{a}}\), finds a number which, when cubed, gives you \(\text{a}\).


Using properties of radicals is helpful for simplifying complex expressions, and rewriting them can make solving easier. In our example, we converted the square root to an exponent form to further simplify the problem.
Exponent Rules
Exponent rules, or laws, allow us to handle powers in algebraic expressions. Here are some key rules that were used in the solution:

  • The rule \(\text{a^{m \cdot n} = (a^m)^n}\) allows multiplication of exponents.
  • The reciprocal rule \(\text{a^{-n} = \frac{1}{a^n}}\) converts a positive exponent in the denominator to a negative exponent in the numerator.


By applying these rules, we managed to convert the fraction containing the square root to a negative exponent. In our exercise:

First, converting the radical expression \(\text{a^{1/2}}\). Then, by applying the reciprocal rule to get the final expression as \( \text{(x^2 + 7)}^{1/2} \rightarrow (x^2 + 7)^{-1/2} \).
Simplifying Expressions
Simplifying expressions reduces complexity, making them easier to analyze or solve.

In our example, we started with \( \frac{1}{\text{something}} \) and transformed it step-by-step into a simpler form. Let's see how we simplified:

First, identify the radical. The given expression was \(\frac{1}{\sqrt{x^{2}+7}}\).
  • Rewrite the square root as an exponent using \( a^{1/2} \) to get \( \frac{1}{(x^2 + 7)^{1/2}} \).
  • Apply the exponent rules to move the term to the numerator as a negative exponent: \((x^2 + 7)^{-1/2} \).


Each step follows logical algebraic manipulations. Recognizing patterns and consistently applying rules simplifies even complex-looking expressions.

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