Algebra Examples

$f (x) = 2 x^{- \frac{2}{3}}$

Step 1

Write $f (x) = 2 x^{- \frac{2}{3}}$ as an equation.

$y = 2 x^{- \frac{2}{3}}$

Step 2

Interchange the variables.

$x = 2 y^{- \frac{2}{3}}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $2 y^{- \frac{2}{3}} = x$ .

$2 y^{- \frac{2}{3}} = x$

Step 3.2

Divide each term in $2 y^{- \frac{2}{3}} = x$ by $2$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $2 y^{- \frac{2}{3}} = x$ by $2$ .

$\frac{2 y^{- \frac{2}{3}}}{2} = \frac{x}{2}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Move $y^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{2 y^{\frac{2}{3}}} = \frac{x}{2}$

Step 3.2.2.2

Cancel the common factor.

$\frac{2}{2 y^{\frac{2}{3}}} = \frac{x}{2}$

Step 3.2.2.3

Rewrite the expression.

$\frac{1}{y^{\frac{2}{3}}} = \frac{x}{2}$

Step 3.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{\frac{2}{3}}, 2$

Step 3.3.2

Since $y^{\frac{2}{3}}, 2$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 2$ then find LCM for the variable part $y^{\frac{2}{3}}$ .

Step 3.3.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3.3.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.3.5

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 3.3.6

The LCM of $1, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 3.3.7

The LCM of $y^{\frac{2}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y^{\frac{2}{3}}$

Step 3.3.8

The LCM for $y^{\frac{2}{3}}, 2$ is the numeric part $2$ multiplied by the variable part.

$2 y^{\frac{2}{3}}$

Step 3.4

Multiply each term in $\frac{1}{y^{\frac{2}{3}}} = \frac{x}{2}$ by $2 y^{\frac{2}{3}}$ to eliminate the fractions.

Tap for more steps...

Step 3.4.1

Multiply each term in $\frac{1}{y^{\frac{2}{3}}} = \frac{x}{2}$ by $2 y^{\frac{2}{3}}$ .

$\frac{1}{y^{\frac{2}{3}}} (2 y^{\frac{2}{3}}) = \frac{x}{2} (2 y^{\frac{2}{3}})$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{1}{y^{\frac{2}{3}}} y^{\frac{2}{3}} = \frac{x}{2} (2 y^{\frac{2}{3}})$

Step 3.4.2.2

Combine $2$ and $\frac{1}{y^{\frac{2}{3}}}$ .

$\frac{2}{y^{\frac{2}{3}}} y^{\frac{2}{3}} = \frac{x}{2} (2 y^{\frac{2}{3}})$

Step 3.4.2.3

Cancel the common factor of $y^{\frac{2}{3}}$ .

Tap for more steps...

Step 3.4.2.3.1

Cancel the common factor.

$\frac{2}{y^{\frac{2}{3}}} y^{\frac{2}{3}} = \frac{x}{2} (2 y^{\frac{2}{3}})$

Step 3.4.2.3.2

Rewrite the expression.

$2 = \frac{x}{2} (2 y^{\frac{2}{3}})$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Rewrite using the commutative property of multiplication.

$2 = 2 \frac{x}{2} y^{\frac{2}{3}}$

Step 3.4.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.3.2.1

Cancel the common factor.

$2 = 2 \frac{x}{2} y^{\frac{2}{3}}$

Step 3.4.3.2.2

Rewrite the expression.

$2 = x y^{\frac{2}{3}}$

Step 3.5

Solve the equation.

Tap for more steps...

Step 3.5.1

Rewrite the equation as $x y^{\frac{2}{3}} = 2$ .

$x y^{\frac{2}{3}} = 2$

Step 3.5.2

Divide each term in $x y^{\frac{2}{3}} = 2$ by $x$ and simplify.

Tap for more steps...

Step 3.5.2.1

Divide each term in $x y^{\frac{2}{3}} = 2$ by $x$ .

$\frac{x y^{\frac{2}{3}}}{x} = \frac{2}{x}$

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Cancel the common factor.

$\frac{x y^{\frac{2}{3}}}{x} = \frac{2}{x}$

Step 3.5.2.2.2

Divide $y^{\frac{2}{3}}$ by $1$ .

$y^{\frac{2}{3}} = \frac{2}{x}$

Step 3.5.3

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Step 3.5.4

Simplify the exponent.

Tap for more steps...

Step 3.5.4.1

Simplify the left side.

Tap for more steps...

Step 3.5.4.1.1

Simplify ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 3.5.4.1.1.1

Multiply the exponents in ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 3.5.4.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Step 3.5.4.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Step 3.5.4.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{3} \cdot 3} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Step 3.5.4.1.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.4.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Step 3.5.4.1.1.1.3.2

Rewrite the expression.

$y^{1} = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Step 3.5.4.1.1.2

Simplify.

$y = \pm {(\frac{2}{x})}^{\frac{3}{2}}$

Step 3.5.4.2

Simplify the right side.

Tap for more steps...

Step 3.5.4.2.1

Apply the product rule to $\frac{2}{x}$ .

$y = \pm \frac{2^{\frac{3}{2}}}{x^{\frac{3}{2}}}$

Step 3.5.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{2^{\frac{3}{2}}}{x^{\frac{3}{2}}}$

Step 3.5.5.2

Next, use the negative value of the $\pm$ to find the second solution.