Algebra Examples

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

Step 1

Divide each term in $- 4 {(3 x)}^{\frac{1}{2}} = - 12$ by $- 4$ and simplify.

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Step 1.1

Divide each term in $- 4 {(3 x)}^{\frac{1}{2}} = - 12$ by $- 4$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Apply the product rule to $3 x$ .

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $- 12$ by $- 4$ .

$3^{\frac{1}{2}} x^{\frac{1}{2}} = 3$

Step 2

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

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Step 2.1

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

$\frac{3^{\frac{1}{2}} x^{\frac{1}{2}}}{3^{\frac{1}{2}}} = \frac{3}{3^{\frac{1}{2}}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor.

$\frac{3^{\frac{1}{2}} x^{\frac{1}{2}}}{3^{\frac{1}{2}}} = \frac{3}{3^{\frac{1}{2}}}$

Step 2.2.2

Divide $x^{\frac{1}{2}}$ by $1$ .

$x^{\frac{1}{2}} = \frac{3}{3^{\frac{1}{2}}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move $3^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 2.3.2

Multiply $3$ by $3^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.2.1

Multiply $3$ by $3^{- \frac{1}{2}}$ .

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Step 2.3.2.1.1

Raise $3$ to the power of $1$ .

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

Step 2.3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3.2.3

Combine the numerators over the common denominator.

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

Step 2.3.2.4

Subtract $1$ from $2$ .

$x^{\frac{1}{2}} = 3^{\frac{1}{2}}$

Step 3

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$x = 3$