Solve (1/a*1/b)^-1=ab

$$( \frac { 1 } { a } \times \frac { 1 } { b } ) ^ { - 1 } = a b$$

Solve for a

$$a\neq 0$$

$$b\neq 0$$

Short Solution Steps

Multiply \frac{1}{a} times \frac{1}{b} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{1}{ab}\right)^{-1}=ab$$

To raise \frac{1}{ab} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-1}}{\left(ab\right)^{-1}}=ab$$

Calculate 1 to the power of -1 and get 1.

$$\frac{1}{\left(ab\right)^{-1}}=ab$$

Expand \left(ab\right)^{-1}.

$$\frac{1}{a^{-1}b^{-1}}=ab$$

Subtract ab from both sides.

$$\frac{1}{a^{-1}b^{-1}}-ab=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply ab times \frac{a^{-1}b^{-1}}{a^{-1}b^{-1}}.

$$\frac{1}{a^{-1}b^{-1}}-\frac{aba^{-1}b^{-1}}{a^{-1}b^{-1}}=0$$

Since \frac{1}{a^{-1}b^{-1}} and \frac{aba^{-1}b^{-1}}{a^{-1}b^{-1}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1-aba^{-1}b^{-1}}{a^{-1}b^{-1}}=0$$

Do the multiplications in 1-aba^{-1}b^{-1}.

$$\frac{1-1}{a^{-1}b^{-1}}=0$$

Do the calculations in 1-1.

$$\frac{0}{a^{-1}b^{-1}}=0$$

Reorder the terms.

$$\frac{0}{\frac{1}{a}\times \frac{1}{b}}=0$$

Multiply \frac{1}{a} times \frac{1}{b} by multiplying numerator times numerator and denominator times denominator.

$$\frac{0}{\frac{1}{ab}}=0$$

Variable a cannot be equal to 0 since division by zero is not defined. Divide 0 by \frac{1}{ab} by multiplying 0 by the reciprocal of \frac{1}{ab}.

$$0ab=0$$

Anything times zero gives zero.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Variable a cannot be equal to 0.

$$a\in \mathrm{R}\setminus 0$$

Solve for b

$$b\neq 0$$

$$a\neq 0$$

Short Solution Steps

Multiply \frac{1}{a} times \frac{1}{b} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{1}{ab}\right)^{-1}=ab$$

To raise \frac{1}{ab} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-1}}{\left(ab\right)^{-1}}=ab$$

Calculate 1 to the power of -1 and get 1.

$$\frac{1}{\left(ab\right)^{-1}}=ab$$

Expand \left(ab\right)^{-1}.

$$\frac{1}{a^{-1}b^{-1}}=ab$$

Subtract ab from both sides.

$$\frac{1}{a^{-1}b^{-1}}-ab=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply ab times \frac{a^{-1}b^{-1}}{a^{-1}b^{-1}}.

$$\frac{1}{a^{-1}b^{-1}}-\frac{aba^{-1}b^{-1}}{a^{-1}b^{-1}}=0$$

Since \frac{1}{a^{-1}b^{-1}} and \frac{aba^{-1}b^{-1}}{a^{-1}b^{-1}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1-aba^{-1}b^{-1}}{a^{-1}b^{-1}}=0$$

Do the multiplications in 1-aba^{-1}b^{-1}.

$$\frac{1-1}{a^{-1}b^{-1}}=0$$

Do the calculations in 1-1.

$$\frac{0}{a^{-1}b^{-1}}=0$$

Reorder the terms.

$$\frac{0}{\frac{1}{a}\times \frac{1}{b}}=0$$

Multiply \frac{1}{a} times \frac{1}{b} by multiplying numerator times numerator and denominator times denominator.

$$\frac{0}{\frac{1}{ab}}=0$$

Variable b cannot be equal to 0 since division by zero is not defined. Divide 0 by \frac{1}{ab} by multiplying 0 by the reciprocal of \frac{1}{ab}.

$$0ab=0$$

Anything times zero gives zero.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{R}$$

Variable b cannot be equal to 0.

$$b\in \mathrm{R}\setminus 0$$