Problem 124 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression.$$x y\left(x^{-1}+y^{-1}\right)^{-1}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x+y}\).

Step 1: Understand the rules of exponents

Recall that the rules of exponents are as follows: \(a^{-n} = \frac{1}{a^n}\) and \((a^n)^{-m} = a^{-(nm)}\). These rules allow us to manipulate terms with exponents to simplify complex expressions.

Step 2: Apply the rules to the expression

Applying the rules to the expression \(x y(x^{-1}+y^{-1})^{-1}\), we get \(x y\left(\frac{1}{x}+\frac{1}{y}\right)^{-1}\). Here, \(x^{-1}\) is rewritten as \(\frac{1}{x}\) and \(y^{-1}\) is rewritten as \(\frac{1}{y}\).

Step 3: Simplify the interior of the parenthesis

Since \(\frac{1}{x}+\frac{1}{y}\) are fractions with different denominators, find the least common denominator (LCD) which is xy. Add the two fractions by converting them to the same denominator: \(\frac{1}{x}+\frac{1}{y}=\frac{y}{xy}+\frac{x}{xy}=\frac{x+y}{xy}\). This changes our expression into \(x y\left(\frac{x+y}{xy}\right)^{-1}\).

Step 4: Simplify the complete expression

We simplify the complete expression with the help of the exponent rule \(a^{-n} = \frac{1}{a^n}\) and get \(x y\left(\frac{xy}{x+y}\right)\). The term inside the brackets flips because of the negative exponent.

Step 5: Cancel mutual terms

Now, we multiply \(x y\) with \(\frac{xy} {x + y}\) and the 'xy' term in the numerator cancels with the 'xy' term in the denominator resulting with 1 in the numerator. Therefore, the simplified form of the expression is \(\frac{1}{x+y}\).