Algebra Examples

$y = \frac{1}{8} x^{\frac{3}{2}} - 8$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{1}{8} x^{\frac{3}{2}} - 8$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{8} x^{\frac{3}{2}} - 8 = 0$ .

$\frac{1}{8} x^{\frac{3}{2}} - 8 = 0$

Step 1.2.2

Add $8$ to both sides of the equation.

$\frac{1}{8} x^{\frac{3}{2}} = 8$

Step 1.2.3

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

${(\frac{1}{8} x^{\frac{3}{2}})}^{\frac{2}{3}} = 8^{\frac{2}{3}}$

Step 1.2.4

Simplify the exponent.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Simplify ${(\frac{1}{8} x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 1.2.4.1.1.1

Combine fractions.

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Step 1.2.4.1.1.1.1

Combine $\frac{1}{8}$ and $x^{\frac{3}{2}}$ .

${(\frac{x^{\frac{3}{2}}}{8})}^{\frac{2}{3}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.1.2

Apply the product rule to $\frac{x^{\frac{3}{2}}}{8}$ .

$\frac{{(x^{\frac{3}{2}})}^{\frac{2}{3}}}{8^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.2

Simplify the numerator.

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Step 1.2.4.1.1.2.1

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

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Step 1.2.4.1.1.2.1.1

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

$\frac{x^{\frac{3}{2} \cdot \frac{2}{3}}}{8^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.2.1.2

Cancel the common factor of $3$ .

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Step 1.2.4.1.1.2.1.2.1

Cancel the common factor.

$\frac{x^{\frac{3}{2} \cdot \frac{2}{3}}}{8^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.2.1.2.2

Rewrite the expression.

$\frac{x^{\frac{1}{2} \cdot 2}}{8^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.2.1.3

Cancel the common factor of $2$ .

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Step 1.2.4.1.1.2.1.3.1

Cancel the common factor.

$\frac{x^{\frac{1}{2} \cdot 2}}{8^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.2.1.3.2

Rewrite the expression.

$\frac{x^{1}}{8^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.2.2

Simplify.

$\frac{x}{8^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.3

Simplify the denominator.

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Step 1.2.4.1.1.3.1

Rewrite $8$ as $2^{3}$ .

$\frac{x}{{(2^{3})}^{\frac{2}{3}}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.3.2

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

$\frac{x}{2^{3 (\frac{2}{3})}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.3.3

Cancel the common factor of $3$ .

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Step 1.2.4.1.1.3.3.1

Cancel the common factor.

$\frac{x}{2^{3 (\frac{2}{3})}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.3.3.2

Rewrite the expression.

$\frac{x}{2^{2}} = 8^{\frac{2}{3}}$

Step 1.2.4.1.1.3.4

Raise $2$ to the power of $2$ .

$\frac{x}{4} = 8^{\frac{2}{3}}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify $8^{\frac{2}{3}}$ .

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Step 1.2.4.2.1.1

Simplify the expression.

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Step 1.2.4.2.1.1.1

Rewrite $8$ as $2^{3}$ .

$\frac{x}{4} = {(2^{3})}^{\frac{2}{3}}$

Step 1.2.4.2.1.1.2

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

$\frac{x}{4} = 2^{3 (\frac{2}{3})}$

Step 1.2.4.2.1.2

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.2.1

Cancel the common factor.

$\frac{x}{4} = 2^{3 (\frac{2}{3})}$

Step 1.2.4.2.1.2.2

Rewrite the expression.

$\frac{x}{4} = 2^{2}$

Step 1.2.4.2.1.3

Raise $2$ to the power of $2$ .

$\frac{x}{4} = 4$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Multiply both sides of the equation by $4$ .

$4 \frac{x}{4} = 4 \cdot 4$

Step 1.2.5.2

Simplify both sides of the equation.

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Step 1.2.5.2.1

Simplify the left side.

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Step 1.2.5.2.1.1

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 \cdot 4$

Step 1.2.5.2.1.1.2

Rewrite the expression.

$x = 4 \cdot 4$

Step 1.2.5.2.2

Simplify the right side.

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Step 1.2.5.2.2.1

Multiply $4$ by $4$ .

$x = 16$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(16, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{1}{8} \cdot {(0)}^{\frac{3}{2}} - 8$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{8} \cdot 0^{\frac{3}{2}} - 8$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{8} \cdot {(0)}^{\frac{3}{2}} - 8$

Step 2.2.3

Simplify $\frac{1}{8} \cdot {(0)}^{\frac{3}{2}} - 8$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite $0$ as $0^{2}$ .

$y = \frac{1}{8} \cdot {(0^{2})}^{\frac{3}{2}} - 8$

Step 2.2.3.1.2

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

$y = \frac{1}{8} \cdot 0^{2 (\frac{3}{2})} - 8$

Step 2.2.3.1.3

Cancel the common factor of $2$ .

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Step 2.2.3.1.3.1

Cancel the common factor.

$y = \frac{1}{8} \cdot 0^{2 (\frac{3}{2})} - 8$

Step 2.2.3.1.3.2

Rewrite the expression.

$y = \frac{1}{8} \cdot 0^{3} - 8$

Step 2.2.3.1.4

Raising $0$ to any positive power yields $0$ .

$y = \frac{1}{8} \cdot 0 - 8$

Step 2.2.3.1.5

Multiply $\frac{1}{8}$ by $0$ .

$y = 0 - 8$

Step 2.2.3.2

Subtract $8$ from $0$ .

$y = - 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 8)$

Step 3

List the intersections.

x-intercept(s): $(16, 0)$

y-intercept(s): $(0, - 8)$

Step 4