Algebra Examples

$\frac{\sqrt{4 - 3 x}}{\sqrt{3 x}} = 2$

Step 1

Cross multiply.

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Step 1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$2 \cdot (\sqrt{3 x}) = \sqrt{4 - 3 x}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Multiply $2$ by $\sqrt{3 x}$ .

$2 \sqrt{3 x} = \sqrt{4 - 3 x}$

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

${(2 {(3 x)}^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $3 x$ .

${(2 (3^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.2

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

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Step 3.2.1.2.1

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

${(2 \cdot 3^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.2.2

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

$2^{2} \cdot {(3^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.3

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

$4 \cdot {(3^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

$4 \cdot 3^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

$4 \cdot 3^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.4.2.2

Rewrite the expression.

$4 \cdot 3^{1} {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.5

Evaluate the exponent.

$4 \cdot 3 {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.6

Multiply $4$ by $3$ .

$12 {(x^{\frac{1}{2}})}^{2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.7

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

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Step 3.2.1.7.1

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

$12 x^{\frac{1}{2} \cdot 2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.7.2

Cancel the common factor of $2$ .

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Step 3.2.1.7.2.1

Cancel the common factor.

$12 x^{\frac{1}{2} \cdot 2} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.7.2.2

Rewrite the expression.

$12 x^{1} = {\sqrt{4 - 3 x}}^{2}$

Step 3.2.1.8

Simplify.

$12 x = {\sqrt{4 - 3 x}}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Rewrite ${\sqrt{4 - 3 x}}^{2}$ as $4 - 3 x$ .

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Step 3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - 3 x}$ as ${(4 - 3 x)}^{\frac{1}{2}}$ .

$12 x = {({(4 - 3 x)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.2

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

$12 x = {(4 - 3 x)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$12 x = {(4 - 3 x)}^{\frac{2}{2}}$

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

Cancel the common factor.

$12 x = {(4 - 3 x)}^{\frac{2}{2}}$

Step 3.3.1.4.2

Rewrite the expression.

$12 x = {(4 - 3 x)}^{1}$

Step 3.3.1.5

Simplify.

$12 x = 4 - 3 x$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Add $3 x$ to both sides of the equation.

$12 x + 3 x = 4$

Step 4.1.2

Add $12 x$ and $3 x$ .

$15 x = 4$

Step 4.2

Divide each term in $15 x = 4$ by $15$ and simplify.

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Step 4.2.1

Divide each term in $15 x = 4$ by $15$ .

$\frac{15 x}{15} = \frac{4}{15}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $15$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} = \frac{4}{15}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{15}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{15}$

Decimal Form:

$x = 0.2\overline{6}$