Finite Math Examples

7 x^{\frac{2}{3}} - 252 = 0

$7 x^{\frac{2}{3}} - 252 = 0$

Step 1

Add

252

$252$ to both sides of the equation.

7 x^{\frac{2}{3}} = 252

$7 x^{\frac{2}{3}} = 252$

Step 2

Raise each side of the equation to the power of

\frac{3}{2}

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

{(7 x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 252^{\frac{3}{2}}

${(7 x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 252^{\frac{3}{2}}$

Step 3

Simplify the left side.

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Step 3.1

Simplify

{(7 x^{\frac{2}{3}})}^{\frac{3}{2}}

${(7 x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.1.1

Apply the product rule to

7 x^{\frac{2}{3}}

$7 x^{\frac{2}{3}}$ .

7^{\frac{3}{2}} {(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 252^{\frac{3}{2}}

$7^{\frac{3}{2}} {(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 252^{\frac{3}{2}}$

Step 3.1.2

Multiply the exponents in

{(x^{\frac{2}{3}})}^{\frac{3}{2}}

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

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Step 3.1.2.1

Apply the power rule and multiply exponents,

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

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

7^{\frac{3}{2}} x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 252^{\frac{3}{2}}

$7^{\frac{3}{2}} x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 252^{\frac{3}{2}}$

Step 3.1.2.2

Cancel the common factor of

2

$2$ .

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Step 3.1.2.2.1

Cancel the common factor.

$7^{\frac{3}{2}} x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 252^{\frac{3}{2}}$

Step 3.1.2.2.2

Rewrite the expression.

$7^{\frac{3}{2}} x^{\frac{1}{3} \cdot 3} = \pm 252^{\frac{3}{2}}$

Step 3.1.2.3

Cancel the common factor of

$3$ .

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Step 3.1.2.3.1

Cancel the common factor.

$7^{\frac{3}{2}} x^{\frac{1}{3} \cdot 3} = \pm 252^{\frac{3}{2}}$

Step 3.1.2.3.2

Rewrite the expression.

$7^{\frac{3}{2}} x^{1} = \pm 252^{\frac{3}{2}}$

Step 3.1.3

Simplify.

$7^{\frac{3}{2}} x = \pm 252^{\frac{3}{2}}$

Step 3.1.4

Reorder factors in

$7^{\frac{3}{2}} x$ .

$x \cdot 7^{\frac{3}{2}} = \pm 252^{\frac{3}{2}}$

Step 4

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

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Step 4.1

First, use the positive value of the

$\pm$ to find the first solution.

$x \cdot 7^{\frac{3}{2}} = 252^{\frac{3}{2}}$

Step 4.2

Divide each term in

$x \cdot 7^{\frac{3}{2}} = 252^{\frac{3}{2}}$ by

$7^{\frac{3}{2}}$ and simplify.

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Step 4.2.1

Divide each term in

$x \cdot 7^{\frac{3}{2}} = 252^{\frac{3}{2}}$ by

$7^{\frac{3}{2}}$ .

$\frac{x \cdot 7^{\frac{3}{2}}}{7^{\frac{3}{2}}} = \frac{252^{\frac{3}{2}}}{7^{\frac{3}{2}}}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor.

$\frac{x \cdot 7^{\frac{3}{2}}}{7^{\frac{3}{2}}} = \frac{252^{\frac{3}{2}}}{7^{\frac{3}{2}}}$

Step 4.2.2.2

Divide

$x$ by

$1$ .

$x = \frac{252^{\frac{3}{2}}}{7^{\frac{3}{2}}}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Use the power of quotient rule

$\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$x = {(\frac{252}{7})}^{\frac{3}{2}}$

Step 4.2.3.2

Simplify the expression.

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Step 4.2.3.2.1

Divide

$252$ by

$7$ .

$x = 36^{\frac{3}{2}}$

Step 4.2.3.2.2

Rewrite

$36$ as

$6^{2}$ .

$x = {(6^{2})}^{\frac{3}{2}}$

Step 4.2.3.2.3

Apply the power rule and multiply exponents,

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

$x = 6^{2 (\frac{3}{2})}$

Step 4.2.3.3

Cancel the common factor of

$2$ .

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Step 4.2.3.3.1

Cancel the common factor.

$x = 6^{2 (\frac{3}{2})}$

Step 4.2.3.3.2

Rewrite the expression.

$x = 6^{3}$

Step 4.2.3.4

Raise

$6$ to the power of

$3$ .

$x = 216$

Step 4.3

Next, use the negative value of the

$\pm$ to find the second solution.

$x \cdot 7^{\frac{3}{2}} = - 252^{\frac{3}{2}}$

Step 4.4

Divide each term in

$x \cdot 7^{\frac{3}{2}} = - 252^{\frac{3}{2}}$ by

$7^{\frac{3}{2}}$ and simplify.

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Step 4.4.1

Divide each term in

$x \cdot 7^{\frac{3}{2}} = - 252^{\frac{3}{2}}$ by

$7^{\frac{3}{2}}$ .

$\frac{x \cdot 7^{\frac{3}{2}}}{7^{\frac{3}{2}}} = \frac{- 252^{\frac{3}{2}}}{7^{\frac{3}{2}}}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor.

$\frac{x \cdot 7^{\frac{3}{2}}}{7^{\frac{3}{2}}} = \frac{- 252^{\frac{3}{2}}}{7^{\frac{3}{2}}}$

Step 4.4.2.2

Divide

$x$ by

$1$ .

$x = \frac{- 252^{\frac{3}{2}}}{7^{\frac{3}{2}}}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Move the negative in front of the fraction.

$x = - \frac{252^{\frac{3}{2}}}{7^{\frac{3}{2}}}$

Step 4.4.3.2

Use the power of quotient rule

$\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$x = - {(\frac{252}{7})}^{\frac{3}{2}}$

Step 4.4.3.3

Simplify the expression.

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Step 4.4.3.3.1

Divide

$252$ by

$7$ .

$x = - 36^{\frac{3}{2}}$

Step 4.4.3.3.2

Rewrite

$36$ as

$6^{2}$ .

$x = - {(6^{2})}^{\frac{3}{2}}$

Step 4.4.3.3.3

Apply the power rule and multiply exponents,

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

$x = - 6^{2 (\frac{3}{2})}$

Step 4.4.3.4

Cancel the common factor of

$2$ .

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Step 4.4.3.4.1

Cancel the common factor.

$x = - 6^{2 (\frac{3}{2})}$

Step 4.4.3.4.2

Rewrite the expression.

$x = - 6^{3}$

Step 4.4.3.5

Simplify the expression.

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Step 4.4.3.5.1

Raise

$6$ to the power of

$3$ .

$x = - 1 \cdot 216$

Step 4.4.3.5.2

Multiply

$- 1$ by

$216$ .

$x = - 216$

Step 4.5

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

$x = 216, - 216$

Step 5