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Chapter 7: Problem 44

Find the reciprocal of each expression. $$\frac{1}{a^{2}-b^{2}}$$

Short Answer

Expert verified
The reciprocal is \(a^{2} - b^{2}\).

Step by step solution

01

Understand the Reciprocal

The reciprocal of a fraction \(\frac{p}{q}\) is given by \(\frac{q}{p}\). To solve for the reciprocal of the given expression \(\frac{1}{a^{2}-b^{2}}\), first identify the numerator and the denominator.
02

Identify the Fraction Components

In this expression, the numerator is 1 and the denominator is \(a^{2} - b^{2}\).
03

Find the Reciprocal

Flip the fraction to find the reciprocal. When you do this, the numerator and the denominator switch places. The reciprocal of \(\frac{1}{a^{2}-b^{2}}\) is \(a^{2} - b^{2}\).

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Key Concepts

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

Rational Expressions
In algebra, a rational expression is a fraction where both the numerator and the denominator are polynomials. Rational expressions are similar to regular fractions, but they contain variables. For example, \(\frac{2x+3}{x^2-1}\) is a rational expression. These expressions can be simplified, added, subtracted, multiplied, and divided just like numerical fractions. However, they also have some extra rules to follow when dealing with variable restrictions.
Fraction Components
Every fraction has two main components: the numerator and the denominator. The numerator is the top part of the fraction, while the denominator is the bottom part. In a fraction \(\frac{a}{b}\), 'a' is the numerator and 'b' is the denominator. The components in a rational expression can be more complex due to the presence of variables and exponents. In our example, \(\frac{1}{a^2 - b^2}\), '1' is the numerator and \(a^2 - b^2\) is the denominator.
Reciprocals
A reciprocal is a flipped fraction. The reciprocal of a fraction \(\frac{p}{q}\) is \(\frac{q}{p}\). To find the reciprocal, swap the positions of the numerator and the denominator. For instance, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). For rational expressions, the process is the same. If you have \(\frac{1}{a^2 - b^2}\), its reciprocal would be \(a^2 - b^2\). This technique is useful in many algebraic manipulations, including simplifying complex fractions and solving equations.

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Most popular questions from this chapter

Perform the indicated operations. Simplify, if possible. $$\frac{x+y}{x^{2}-y^{2}}+\frac{x-y}{x^{2}-y^{2}}-\frac{2 x}{x^{2}-y^{2}}$$

Solve. $$1+\frac{x-1}{x-3}=\frac{2}{x-3}-x$$

ListAndSell is an auction dropoff store that accepts items for sale on Internet auctions. For a fee, the items are listed on the auction and shipped when sold. The table below lists the amount that a seller receives from the sale of various items. $$\begin{array}{l|c}{\text { Item Selling Price }} & {\text { Amount Seller Receives }} \\ {\$ 75.00} & {\$ 41.42} \\ {100.00} & {55.50} \\ {200.00} & {111.85} \\ {400.00} & {240.55}\end{array}$$ a) Determine whether the amount that the seller receives varies directly or inversely as the selling price. b) Find an equation of variation that approximates the data. Use the data point \((200,111.85)\) c) Use the equation to predict the amount the seller receives if an item sells for \(\$ 150.00 .\)

Find an equation of variation in which: \(y\) varies directly as the square of \(x,\) and \(y=0.15\) when \(x=0.1\)

The tension \(T\) on a string in a musical instrument varies jointly as the string's mass per unit length \(m,\) the square of its length \(l,\) and the square of its fundamental frequency \(f . \mathrm{A} 2-\mathrm{m}\) long string of mass \(5 \mathrm{gm} / \mathrm{m}\) with a fundamental frequency of 80 has a tension of \(100 \mathrm{N}\). How long should the same string be if its tension is going to be changed to \(72 \mathrm{N} ?\)
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