Simplify (5x^-1+2y^-1)/(x^-2y^-2)

$\frac{5 x^{- 1} + 2 y^{- 1}}{x^{- 2} y^{- 2}}$

Step 1

Move $x^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{(5 x^{- 1} + 2 y^{- 1}) x^{2}}{y^{- 2}}$

Step 2

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$(5 x^{- 1} + 2 y^{- 1}) x^{2} y^{2}$

Step 3

Simplify each term.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(5 \frac{1}{x} + 2 y^{- 1}) x^{2} y^{2}$

Combine $5$ and $\frac{1}{x}$ .

$(\frac{5}{x} + 2 y^{- 1}) x^{2} y^{2}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(\frac{5}{x} + 2 \frac{1}{y}) x^{2} y^{2}$

Combine $2$ and $\frac{1}{y}$ .

$(\frac{5}{x} + \frac{2}{y}) x^{2} y^{2}$

Step 4

To write $\frac{5}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$(\frac{5}{x} \cdot \frac{y}{y} + \frac{2}{y}) x^{2} y^{2}$

Step 5

To write $\frac{2}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$(\frac{5}{x} \cdot \frac{y}{y} + \frac{2}{y} \cdot \frac{x}{x}) x^{2} y^{2}$

Step 6

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{x}$ by $\frac{y}{y}$ .

$(\frac{5 y}{x y} + \frac{2}{y} \cdot \frac{x}{x}) x^{2} y^{2}$

Multiply $\frac{2}{y}$ by $\frac{x}{x}$ .

$(\frac{5 y}{x y} + \frac{2 x}{y x}) x^{2} y^{2}$

Reorder the factors of $y x$ .

$(\frac{5 y}{x y} + \frac{2 x}{x y}) x^{2} y^{2}$

Step 7

Combine the numerators over the common denominator.

$\frac{5 y + 2 x}{x y} x^{2} y^{2}$

Step 8

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

$\frac{5 y + 2 x}{x (y)} x^{2} y^{2}$

Factor $x$ out of $x^{2}$ .

$\frac{5 y + 2 x}{x (y)} (x \cdot x) y^{2}$

Cancel the common factor.

$\frac{5 y + 2 x}{x y} (x \cdot x) y^{2}$

Rewrite the expression.

$\frac{5 y + 2 x}{y} x y^{2}$

Step 9

Combine $\frac{5 y + 2 x}{y}$ and $x$ .

$\frac{(5 y + 2 x) x}{y} y^{2}$

Step 10

Cancel the common factor of $y$ .

Factor $y$ out of $y^{2}$ .

$\frac{(5 y + 2 x) x}{y} (y \cdot y)$

Cancel the common factor.

$\frac{(5 y + 2 x) x}{y} (y \cdot y)$

Rewrite the expression.

$(5 y + 2 x) x y$

Step 11

Apply the distributive property.

$(5 y x + 2 x \cdot x) y$

Step 12

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(5 y x + 2 (x \cdot x)) y$

Multiply $x$ by $x$ .

$(5 y x + 2 x^{2}) y$

Step 13

Apply the distributive property.

$5 y x y + 2 x^{2} y$

Step 14

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$5 (y \cdot y) x + 2 x^{2} y$

Multiply $y$ by $y$ .

$5 y^{2} x + 2 x^{2} y$

Simplify (5x^-1+2y^-1)/(x^-2y^-2)

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