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Problem 76

# Write the expression $$\left(\mathrm{x}+\mathrm{y}^{-1}\right)^{-1}$$ without using negative exponents.

Expert verified
The simplified expression without negative exponents is $$\frac{\mathrm{y}}{\mathrm{xy}+1}$$.
See the step by step solution

## Step 1: Rewrite the expression with positive exponents

Since $$\mathrm{y}^{-1} = \frac{1}{\mathrm{y}}$$, we can rewrite the expression as: $$\left(\mathrm{x} + \frac{1}{\mathrm{y}}\right)^{-1}$$ Now, we need to handle the outer exponent -1.

## Step 2: Remove the outer exponent -1

To remove the outer exponent -1, we can use the reciprocal, which means we will take the reciprocal of the whole expression inside the parentheses. So, $$\left(\mathrm{x} + \frac{1}{\mathrm{y}}\right)^{-1} = \frac{1}{\left(\mathrm{x} + \frac{1}{\mathrm{y}}\right)}$$

## Step 3: Simplify the expression

In order to simplify the expression further, we will make the denominators in the expression the same. This will allow us to combine the terms in the denominator. To do this, we will multiply $$\mathrm{x}$$ by $$\frac{\mathrm{y}}{\mathrm{y}}$$ (which equals to 1): $$\frac{1}{\left(\mathrm{x} + \frac{1}{\mathrm{y}}\right)} = \frac{1}{\left(\frac{\mathrm{xy}}{\mathrm{y}} + \frac{1}{\mathrm{y}}\right)}$$ Now, we can combine the terms in the denominator since they have the same denominator ($$\mathrm{y}$$): $$\frac{1}{\left(\frac{\mathrm{xy}}{\mathrm{y}} + \frac{1}{\mathrm{y}}\right)} = \frac{1}{\frac{\mathrm{xy}+1}{\mathrm{y}}}$$ Lastly, we can remove the complex fraction by multiplying the numerator and denominator with $$\mathrm{y}$$: $$\frac{1}{\frac{\mathrm{xy}+1}{\mathrm{y}}} = \frac{1 \cdot \mathrm{y}}{(\mathrm{xy}+1)}$$ So the final expression without negative exponents is: $$\frac{\mathrm{y}}{\mathrm{xy}+1}$$

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