Calculus Examples

$15 - \frac{3}{20} x^{\frac{1}{2}} = 0$

Step 1

Subtract $15$ from both sides of the equation.

$- \frac{3}{20} x^{\frac{1}{2}} = - 15$

Step 2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(- \frac{3}{20} x^{\frac{1}{2}})}^{2} = {(- 15)}^{2}$

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify ${(- \frac{3}{20} x^{\frac{1}{2}})}^{2}$ .

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Step 3.1.1.1

Combine $x^{\frac{1}{2}}$ and $\frac{3}{20}$ .

${(- \frac{x^{\frac{1}{2}} \cdot 3}{20})}^{2} = {(- 15)}^{2}$

Step 3.1.1.2

Move $3$ to the left of $x^{\frac{1}{2}}$ .

${(- \frac{3 \cdot x^{\frac{1}{2}}}{20})}^{2} = {(- 15)}^{2}$

Step 3.1.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.1.3.1

Apply the product rule to $- \frac{3 x^{\frac{1}{2}}}{20}$ .

${(- 1)}^{2} {(\frac{3 x^{\frac{1}{2}}}{20})}^{2} = {(- 15)}^{2}$

Step 3.1.1.3.2

Apply the product rule to $\frac{3 x^{\frac{1}{2}}}{20}$ .

${(- 1)}^{2} \frac{{(3 x^{\frac{1}{2}})}^{2}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.3.3

Apply the product rule to $3 x^{\frac{1}{2}}$ .

${(- 1)}^{2} \frac{3^{2} {(x^{\frac{1}{2}})}^{2}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.4

Simplify the expression.

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Step 3.1.1.4.1

Raise $- 1$ to the power of $2$ .

$1 \frac{3^{2} {(x^{\frac{1}{2}})}^{2}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.4.2

Multiply $\frac{3^{2} {(x^{\frac{1}{2}})}^{2}}{20^{2}}$ by $1$ .

$\frac{3^{2} {(x^{\frac{1}{2}})}^{2}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.5

Simplify the numerator.

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Step 3.1.1.5.1

Raise $3$ to the power of $2$ .

$\frac{9 {(x^{\frac{1}{2}})}^{2}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.5.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.1.1.5.2.1

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

$\frac{9 x^{\frac{1}{2} \cdot 2}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.5.2.2

Cancel the common factor of $2$ .

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Step 3.1.1.5.2.2.1

Cancel the common factor.

$\frac{9 x^{\frac{1}{2} \cdot 2}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.5.2.2.2

Rewrite the expression.

$\frac{9 x^{1}}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.5.3

Simplify.

$\frac{9 x}{20^{2}} = {(- 15)}^{2}$

Step 3.1.1.6

Raise $20$ to the power of $2$ .

$\frac{9 x}{400} = {(- 15)}^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Raise $- 15$ to the power of $2$ .

$\frac{9 x}{400} = 225$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $\frac{400}{9}$ .

$\frac{400}{9} \cdot \frac{9 x}{400} = \frac{400}{9} \cdot 225$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $\frac{400}{9} \cdot \frac{9 x}{400}$ .

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Step 4.2.1.1.1

Cancel the common factor of $400$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

$\frac{400}{9} \cdot \frac{9 x}{400} = \frac{400}{9} \cdot 225$

Step 4.2.1.1.1.2

Rewrite the expression.

$\frac{1}{9} (9 x) = \frac{400}{9} \cdot 225$

Step 4.2.1.1.2

Cancel the common factor of $9$ .

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Step 4.2.1.1.2.1

Factor $9$ out of $9 x$ .

$\frac{1}{9} (9 (x)) = \frac{400}{9} \cdot 225$

Step 4.2.1.1.2.2

Cancel the common factor.

$\frac{1}{9} (9 x) = \frac{400}{9} \cdot 225$

Step 4.2.1.1.2.3

Rewrite the expression.

$x = \frac{400}{9} \cdot 225$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $\frac{400}{9} \cdot 225$ .

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Step 4.2.2.1.1

Cancel the common factor of $9$ .

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Step 4.2.2.1.1.1

Factor $9$ out of $225$ .

$x = \frac{400}{9} \cdot (9 (25))$

Step 4.2.2.1.1.2

Cancel the common factor.

$x = \frac{400}{9} \cdot (9 \cdot 25)$

Step 4.2.2.1.1.3

Rewrite the expression.

$x = 400 \cdot 25$

Step 4.2.2.1.2

Multiply $400$ by $25$ .

$x = 10000$