Algebra Examples

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

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 2

Solve the equation for $x$ .

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Step 2.1

Rewrite the equation as $2 {(3 x - 3)}^{\frac{1}{2}} \cdot 1 = 3 \cdot 3$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Multiply $2$ by $1$ .

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

Step 2.2.2

Multiply $3$ by $3$ .

$2 {(3 x - 3)}^{\frac{1}{2}} = 9$

Step 2.3

Divide each term in $2 {(3 x - 3)}^{\frac{1}{2}} = 9$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 {(3 x - 3)}^{\frac{1}{2}} = 9$ by $2$ .

$\frac{2 {(3 x - 3)}^{\frac{1}{2}}}{2} = \frac{9}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor.

$\frac{2 {(3 x - 3)}^{\frac{1}{2}}}{2} = \frac{9}{2}$

Step 2.3.2.2

Divide ${(3 x - 3)}^{\frac{1}{2}}$ by $1$ .

${(3 x - 3)}^{\frac{1}{2}} = \frac{9}{2}$

Step 2.4

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

${({(3 x - 3)}^{\frac{1}{2}})}^{2} = {(\frac{9}{2})}^{2}$

Step 2.5

Simplify the exponent.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

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

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Step 2.5.1.1.1

Multiply the exponents in ${({(3 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.5.1.1.1.1

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

${(3 x - 3)}^{\frac{1}{2} \cdot 2} = {(\frac{9}{2})}^{2}$

Step 2.5.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.5.1.1.1.2.1

Cancel the common factor.

${(3 x - 3)}^{\frac{1}{2} \cdot 2} = {(\frac{9}{2})}^{2}$

Step 2.5.1.1.1.2.2

Rewrite the expression.

${(3 x - 3)}^{1} = {(\frac{9}{2})}^{2}$

Step 2.5.1.1.2

Simplify.

$3 x - 3 = {(\frac{9}{2})}^{2}$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify ${(\frac{9}{2})}^{2}$ .

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Step 2.5.2.1.1

Apply the product rule to $\frac{9}{2}$ .

$3 x - 3 = \frac{9^{2}}{2^{2}}$

Step 2.5.2.1.2

Raise $9$ to the power of $2$ .

$3 x - 3 = \frac{81}{2^{2}}$

Step 2.5.2.1.3

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

$3 x - 3 = \frac{81}{4}$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Move all terms not containing $x$ to the right side of the equation.

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Step 2.6.1.1

Add $3$ to both sides of the equation.

$3 x = \frac{81}{4} + 3$

Step 2.6.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$3 x = \frac{81}{4} + 3 \cdot \frac{4}{4}$

Step 2.6.1.3

Combine $3$ and $\frac{4}{4}$ .

$3 x = \frac{81}{4} + \frac{3 \cdot 4}{4}$

Step 2.6.1.4

Combine the numerators over the common denominator.

$3 x = \frac{81 + 3 \cdot 4}{4}$

Step 2.6.1.5

Simplify the numerator.

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Step 2.6.1.5.1

Multiply $3$ by $4$ .

$3 x = \frac{81 + 12}{4}$

Step 2.6.1.5.2

Add $81$ and $12$ .

$3 x = \frac{93}{4}$

Step 2.6.2

Divide each term in $3 x = \frac{93}{4}$ by $3$ and simplify.

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Step 2.6.2.1

Divide each term in $3 x = \frac{93}{4}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{93}{4}}{3}$

Step 2.6.2.2

Simplify the left side.

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Step 2.6.2.2.1

Cancel the common factor of $3$ .

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Step 2.6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{93}{4}}{3}$

Step 2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{93}{4}}{3}$

Step 2.6.2.3

Simplify the right side.

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Step 2.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{93}{4} \cdot \frac{1}{3}$

Step 2.6.2.3.2

Cancel the common factor of $3$ .

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Step 2.6.2.3.2.1

Factor $3$ out of $93$ .

$x = \frac{3 (31)}{4} \cdot \frac{1}{3}$

Step 2.6.2.3.2.2

Cancel the common factor.

$x = \frac{3 \cdot 31}{4} \cdot \frac{1}{3}$

Step 2.6.2.3.2.3

Rewrite the expression.

$x = \frac{31}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{31}{4}$

Decimal Form:

$x = 7.75$

Mixed Number Form:

$x = 7 \frac{3}{4}$