Finite Math Examples

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

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

Step 1

Solve the equation for

y

$y$ .

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Step 1.1

Rewrite the equation as

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

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

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

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

Step 1.2

Raise each side of the equation to the power of

\frac{2}{3}

$\frac{2}{3}$ to eliminate the fractional exponent on the left side.

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

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

Step 1.3

Simplify the left side.

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Step 1.3.1

Simplify

{(\frac{1}{3} \cdot {(y^{2} + 2)}^{\frac{3}{2}})}^{\frac{2}{3}}

${(\frac{1}{3} \cdot {(y^{2} + 2)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 1.3.1.1

Combine fractions.

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Step 1.3.1.1.1

Combine

\frac{1}{3}

$\frac{1}{3}$ and

{(y^{2} + 2)}^{\frac{3}{2}}

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

{⎛ ⎜ ⎝ \frac{{(y^{2} + 2)}^{\frac{3}{2}}}{3} ⎞ ⎟ ⎠}^{\frac{2}{3}} = x^{\frac{2}{3}}

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

Step 1.3.1.1.2

Apply the product rule to

\frac{{(y^{2} + 2)}^{\frac{3}{2}}}{3}

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

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

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

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

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

Step 1.3.1.2

Simplify the numerator.

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Step 1.3.1.2.1

Multiply the exponents in

{({(y^{2} + 2)}^{\frac{3}{2}})}^{\frac{2}{3}}

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

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Step 1.3.1.2.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 1.3.1.2.1.2

Cancel the common factor of

3

$3$ .

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Step 1.3.1.2.1.2.1

Cancel the common factor.

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

Step 1.3.1.2.1.2.2

Rewrite the expression.

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

Step 1.3.1.2.1.3

Cancel the common factor of

$2$ .

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Step 1.3.1.2.1.3.1

Cancel the common factor.

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

Step 1.3.1.2.1.3.2

Rewrite the expression.

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

Step 1.3.1.2.2

Simplify.

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

Step 1.3.1.3

Split the fraction

$\frac{y^{2} + 2}{3^{\frac{2}{3}}}$ into two fractions.

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

Step 1.4

Solve for

$y$ .

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Step 1.4.1

Subtract

$\frac{2}{3^{\frac{2}{3}}}$ from both sides of the equation.

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

Step 1.4.2

Multiply each term in

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

$3^{\frac{2}{3}}$ to eliminate the fractions.

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Step 1.4.2.1

Multiply each term in

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

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

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

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of

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

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Step 1.4.2.2.1.1

Cancel the common factor.

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

Step 1.4.2.2.1.2

Rewrite the expression.

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

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Cancel the common factor of

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

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Step 1.4.2.3.1.1

Move the leading negative in

$- \frac{2}{3^{\frac{2}{3}}}$ into the numerator.

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

Step 1.4.2.3.1.2

Cancel the common factor.

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

Step 1.4.2.3.1.3

Rewrite the expression.

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

Step 1.5

Combine

$\frac{1}{3}$ and

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

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

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be

$0$ or

$1$ for each of its variables. In this case and the degree of variable

$x$ is

$1$ . the degrees of the variables in the equation violate the linear equation definition, which means that the equation is not a linear equation.

Not Linear