Algebra Examples

${(x - 2)}^{\frac{1}{2}} = {(28 - 2 x)}^{\frac{1}{4}}$

Step 1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

${({(x - 2)}^{\frac{1}{2}})}^{4} = {({(28 - 2 x)}^{\frac{1}{4}})}^{4}$

Step 2

Multiply the exponents in ${({(x - 2)}^{\frac{1}{2}})}^{4}$ .

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Step 2.1

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

${(x - 2)}^{\frac{1}{2} \cdot 4} = {({(28 - 2 x)}^{\frac{1}{4}})}^{4}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $4$ .

${(x - 2)}^{\frac{1}{2} \cdot (2 (2))} = {({(28 - 2 x)}^{\frac{1}{4}})}^{4}$

Step 2.2.2

Cancel the common factor.

${(x - 2)}^{\frac{1}{2} \cdot (2 \cdot 2)} = {({(28 - 2 x)}^{\frac{1}{4}})}^{4}$

Step 2.2.3

Rewrite the expression.

${(x - 2)}^{2} = {({(28 - 2 x)}^{\frac{1}{4}})}^{4}$

Step 3

Simplify ${({(28 - 2 x)}^{\frac{1}{4}})}^{4}$ .

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Step 3.1

Multiply the exponents in ${({(28 - 2 x)}^{\frac{1}{4}})}^{4}$ .

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Step 3.1.1

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

${(x - 2)}^{2} = {(28 - 2 x)}^{\frac{1}{4} \cdot 4}$

Step 3.1.2

Cancel the common factor of $4$ .

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Step 3.1.2.1

Cancel the common factor.

${(x - 2)}^{2} = {(28 - 2 x)}^{\frac{1}{4} \cdot 4}$

Step 3.1.2.2

Rewrite the expression.

${(x - 2)}^{2} = {(28 - 2 x)}^{1}$

Step 3.2

Simplify.

${(x - 2)}^{2} = 28 - 2 x$

Step 4

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

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Step 4.1

Add $2 x$ to both sides of the equation.

${(x - 2)}^{2} + 2 x = 28$

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(x - 2) (x - 2) + 2 x = 28$

Step 4.2.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 4.2.2.1

Apply the distributive property.

$x (x - 2) - 2 (x - 2) + 2 x = 28$

Step 4.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2) + 2 x = 28$

Step 4.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + 2 x = 28$

Step 4.2.3

Simplify and combine like terms.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 + 2 x = 28$

Step 4.2.3.1.2

Move $- 2$ to the left of $x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 + 2 x = 28$

Step 4.2.3.1.3

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x - 2 x + 4 + 2 x = 28$

Step 4.2.3.2

Subtract $2 x$ from $- 2 x$ .

$x^{2} - 4 x + 4 + 2 x = 28$

Step 4.3

Add $- 4 x$ and $2 x$ .

$x^{2} - 2 x + 4 = 28$

Step 5

Subtract $28$ from both sides of the equation.

$x^{2} - 2 x + 4 - 28 = 0$

Step 6

Subtract $28$ from $4$ .

$x^{2} - 2 x - 24 = 0$

Step 7

Factor $x^{2} - 2 x - 24$ using the AC method.

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Step 7.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 7.2

Write the factored form using these integers.

$(x - 6) (x + 4) = 0$

Step 8

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 4 = 0$

Step 9

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 9.2

Add $6$ to both sides of the equation.

$x = 6$

Step 10

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 10.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 11

The final solution is all the values that make $(x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

Step 12

Exclude the solutions that do not make ${(x - 2)}^{\frac{1}{2}} = {(28 - 2 x)}^{\frac{1}{4}}$ true.

$x = 6$