Precalculus Examples

$x = \sqrt{t^{3}}$ , $y = t^{2}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = \sqrt{t^{3}}$

Step 2

Rewrite the equation as $\sqrt{t^{3}} = x$ .

$\sqrt{t^{3}} = x$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{t^{3}}}^{2} = x^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t^{3}}$ as $t^{\frac{3}{2}}$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.2.1

Cancel the common factor.

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

Step 4.2.1.2.2

Rewrite the expression.

$t^{3} = x^{2}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \sqrt[3]{x^{2}}$

Step 6

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = {(\sqrt[3]{x^{2}})}^{2}$

Step 7

Remove parentheses.

$y = {\sqrt[3]{x^{2}}}^{2}$

Step 8

Simplify ${\sqrt[3]{x^{2}}}^{2}$ .

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Step 8.1

Rewrite ${\sqrt[3]{x^{2}}}^{2}$ as $\sqrt[3]{{(x^{2})}^{2}}$ .

$y = \sqrt[3]{{(x^{2})}^{2}}$

Step 8.2

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

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Step 8.2.1

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

$y = \sqrt[3]{x^{2 \cdot 2}}$

Step 8.2.2

Multiply $2$ by $2$ .

$y = \sqrt[3]{x^{4}}$

Step 8.3

Factor out $x^{3}$ .

$y = \sqrt[3]{x^{3} x}$

Step 8.4

Pull terms out from under the radical.

$y = x \sqrt[3]{x}$