Algebra Examples

${(12 - x)}^{\frac{1}{2}} = 6 - a$

Step 1

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

${({(12 - x)}^{\frac{1}{2}})}^{2} = {(6 - a)}^{2}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify ${({(12 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(12 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.1.1.1

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

${(12 - x)}^{\frac{1}{2} \cdot 2} = {(6 - a)}^{2}$

Step 2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(12 - x)}^{\frac{1}{2} \cdot 2} = {(6 - a)}^{2}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(12 - x)}^{1} = {(6 - a)}^{2}$

Step 2.1.1.2

Simplify.

$12 - x = {(6 - a)}^{2}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify ${(6 - a)}^{2}$ .

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Step 2.2.1.1

Rewrite ${(6 - a)}^{2}$ as $(6 - a) (6 - a)$ .

$12 - x = (6 - a) (6 - a)$

Step 2.2.1.2

Expand $(6 - a) (6 - a)$ using the FOIL Method.

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Step 2.2.1.2.1

Apply the distributive property.

$12 - x = 6 (6 - a) - a (6 - a)$

Step 2.2.1.2.2

Apply the distributive property.

$12 - x = 6 \cdot 6 + 6 (- a) - a (6 - a)$

Step 2.2.1.2.3

Apply the distributive property.

$12 - x = 6 \cdot 6 + 6 (- a) - a \cdot 6 - a (- a)$

Step 2.2.1.3

Simplify and combine like terms.

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Step 2.2.1.3.1

Simplify each term.

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Step 2.2.1.3.1.1

Multiply $6$ by $6$ .

$12 - x = 36 + 6 (- a) - a \cdot 6 - a (- a)$

Step 2.2.1.3.1.2

Multiply $- 1$ by $6$ .

$12 - x = 36 - 6 a - a \cdot 6 - a (- a)$

Step 2.2.1.3.1.3

Multiply $6$ by $- 1$ .

$12 - x = 36 - 6 a - 6 a - a (- a)$

Step 2.2.1.3.1.4

Rewrite using the commutative property of multiplication.

$12 - x = 36 - 6 a - 6 a - 1 \cdot - 1 a \cdot a$

Step 2.2.1.3.1.5

Multiply $a$ by $a$ by adding the exponents.

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Step 2.2.1.3.1.5.1

Move $a$ .

$12 - x = 36 - 6 a - 6 a - 1 \cdot - 1 (a \cdot a)$

Step 2.2.1.3.1.5.2

Multiply $a$ by $a$ .

$12 - x = 36 - 6 a - 6 a - 1 \cdot - 1 a^{2}$

Step 2.2.1.3.1.6

Multiply $- 1$ by $- 1$ .

$12 - x = 36 - 6 a - 6 a + 1 a^{2}$

Step 2.2.1.3.1.7

Multiply $a^{2}$ by $1$ .

$12 - x = 36 - 6 a - 6 a + a^{2}$

Step 2.2.1.3.2

Subtract $6 a$ from $- 6 a$ .

$12 - x = 36 - 12 a + a^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $12$ from both sides of the equation.

$- x = 36 - 12 a + a^{2} - 12$

Step 3.1.2

Subtract $12$ from $36$ .

$- x = - 12 a + a^{2} + 24$

Step 3.2

Divide each term in $- x = - 12 a + a^{2} + 24$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x = - 12 a + a^{2} + 24$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 12 a}{- 1} + \frac{a^{2}}{- 1} + \frac{24}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 12 a}{- 1} + \frac{a^{2}}{- 1} + \frac{24}{- 1}$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 12 a}{- 1} + \frac{a^{2}}{- 1} + \frac{24}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move the negative one from the denominator of $\frac{- 12 a}{- 1}$ .

$x = - 1 \cdot (- 12 a) + \frac{a^{2}}{- 1} + \frac{24}{- 1}$

Step 3.2.3.1.2

Rewrite $- 1 \cdot (- 12 a)$ as $- (- 12 a)$ .

$x = - (- 12 a) + \frac{a^{2}}{- 1} + \frac{24}{- 1}$

Step 3.2.3.1.3

Multiply $- 12$ by $- 1$ .

$x = 12 a + \frac{a^{2}}{- 1} + \frac{24}{- 1}$

Step 3.2.3.1.4

Move the negative one from the denominator of $\frac{a^{2}}{- 1}$ .

$x = 12 a - 1 \cdot a^{2} + \frac{24}{- 1}$

Step 3.2.3.1.5

Rewrite $- 1 \cdot a^{2}$ as $- a^{2}$ .

$x = 12 a - a^{2} + \frac{24}{- 1}$

Step 3.2.3.1.6

Divide $24$ by $- 1$ .

$x = 12 a - a^{2} - 24$