Chapter A.3, Problem 74E

Solution

Write with positive exponents and simplify.

  $\frac{1}{\left(\frac{1}{x}+\frac{1}{y}\right)}$ .

Given:

The given expression is ${\left(x^{-1}+y^{-1}\right)}^{-1}$ .

Concept Used:

• A positive exponent tells us how many times to multiply a base number
• A negative exponent tells us how many times to divide a base number.
• We can rewrite negative exponents like $x^{-n}as\frac{1}{x^n}$ .

Calculation:

The given expression is ${\left(x^{-1}+y^{-1}\right)}^{-1}$ .

  $\begin{array}{c}{\left(x^{-1}+y^{-1}\right)}^{-1}={\left(\frac{1}{x^1}+\frac{1}{y^1}\right)}^{-1}\\ =\frac{1}{{\left(\frac{1}{x}+\frac{1}{y}\right)}^1}\\ =\frac{1}{\left(\frac{1}{x}+\frac{1}{y}\right)}\end{array}$

Thus by simplifying the expression ${\left(x^{-1}+y^{-1}\right)}^{-1}$ the solution is $\frac{1}{\left(\frac{1}{x}+\frac{1}{y}\right)}$ .