Solve by Factoring x^(3/4)=125

$x^{\frac{3}{4}} = 125$

Step 1

Subtract $125$ from both sides of the equation.

$x^{\frac{3}{4}} - 125 = 0$

Step 2

Rewrite $x^{\frac{3}{4}}$ as ${(x^{\frac{1}{4}})}^{3}$ .

${(x^{\frac{1}{4}})}^{3} - 125 = 0$

Step 3

Rewrite $125$ as $5^{3}$ .

${(x^{\frac{1}{4}})}^{3} - 5^{3} = 0$

Step 4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x^{\frac{1}{4}}$ and $b = 5$ .

$(x^{\frac{1}{4}} - 5) ({(x^{\frac{1}{4}})}^{2} + x^{\frac{1}{4}} \cdot 5 + 5^{2}) = 0$

Step 5

Simplify.

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

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

$(x^{\frac{1}{4}} - 5) (x^{\frac{1}{4} \cdot 2} + x^{\frac{1}{4}} \cdot 5 + 5^{2}) = 0$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$(x^{\frac{1}{4}} - 5) (x^{\frac{1}{2 (2)} \cdot 2} + x^{\frac{1}{4}} \cdot 5 + 5^{2}) = 0$

Cancel the common factor.

$(x^{\frac{1}{4}} - 5) (x^{\frac{1}{2 \cdot 2} \cdot 2} + x^{\frac{1}{4}} \cdot 5 + 5^{2}) = 0$

Rewrite the expression.

$(x^{\frac{1}{4}} - 5) (x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 5 + 5^{2}) = 0$

Move $5$ to the left of $x^{\frac{1}{4}}$ .

$(x^{\frac{1}{4}} - 5) (x^{\frac{1}{2}} + 5 x^{\frac{1}{4}} + 5^{2}) = 0$

Raise $5$ to the power of $2$ .

$(x^{\frac{1}{4}} - 5) (x^{\frac{1}{2}} + 5 x^{\frac{1}{4}} + 25) = 0$

Reorder terms.

$(x^{\frac{1}{4}} - 5) (5 x^{\frac{1}{4}} + x^{\frac{1}{2}} + 25) = 0$

Step 6

Simplify $(x^{\frac{1}{4}} - 5) (5 x^{\frac{1}{4}} + x^{\frac{1}{2}} + 25)$ .

Expand $(x^{\frac{1}{4}} - 5) (5 x^{\frac{1}{4}} + x^{\frac{1}{2}} + 25)$ by multiplying each term in the first expression by each term in the second expression.

$x^{\frac{1}{4}} (5 x^{\frac{1}{4}}) + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$5 x^{\frac{1}{4}} x^{\frac{1}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Multiply $x^{\frac{1}{4}}$ by $x^{\frac{1}{4}}$ by adding the exponents.

Move $x^{\frac{1}{4}}$ .

$5 (x^{\frac{1}{4}} x^{\frac{1}{4}}) + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

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

$5 x^{\frac{1}{4} + \frac{1}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Combine the numerators over the common denominator.

$5 x^{\frac{1 + 1}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Add $1$ and $1$ .

$5 x^{\frac{2}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$5 x^{\frac{2 (1)}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Cancel the common factors.

Factor $2$ out of $4$ .

$5 x^{\frac{2 \cdot 1}{2 \cdot 2}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Cancel the common factor.

$5 x^{\frac{2 \cdot 1}{2 \cdot 2}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Rewrite the expression.

$5 x^{\frac{1}{2}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Multiply $x^{\frac{1}{4}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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

$5 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{1}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$5 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{2}{2 \cdot 2}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Multiply $2$ by $2$ .

$5 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{2}{4}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Combine the numerators over the common denominator.

$5 x^{\frac{1}{2}} + x^{\frac{1 + 2}{4}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Add $1$ and $2$ .

$5 x^{\frac{1}{2}} + x^{\frac{3}{4}} + x^{\frac{1}{4}} \cdot 25 - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Move $25$ to the left of $x^{\frac{1}{4}}$ .

$5 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 25 \cdot x^{\frac{1}{4}} - 5 (5 x^{\frac{1}{4}}) - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Multiply $5$ by $- 5$ .

$5 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 25 x^{\frac{1}{4}} - 25 x^{\frac{1}{4}} - 5 x^{\frac{1}{2}} - 5 \cdot 25 = 0$

Multiply $- 5$ by $25$ .

$5 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 25 x^{\frac{1}{4}} - 25 x^{\frac{1}{4}} - 5 x^{\frac{1}{2}} - 125 = 0$

Combine the opposite terms in $5 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 25 x^{\frac{1}{4}} - 25 x^{\frac{1}{4}} - 5 x^{\frac{1}{2}} - 125$ .

Subtract $25 x^{\frac{1}{4}}$ from $25 x^{\frac{1}{4}}$ .

$5 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 0 - 5 x^{\frac{1}{2}} - 125 = 0$

Add $5 x^{\frac{1}{2}} + x^{\frac{3}{4}}$ and $0$ .

$5 x^{\frac{1}{2}} + x^{\frac{3}{4}} - 5 x^{\frac{1}{2}} - 125 = 0$

Subtract $5 x^{\frac{1}{2}}$ from $5 x^{\frac{1}{2}}$ .

$x^{\frac{3}{4}} + 0 - 125 = 0$

Add $x^{\frac{3}{4}}$ and $0$ .

$x^{\frac{3}{4}} - 125 = 0$

Step 7

Add $125$ to both sides of the equation.

$x^{\frac{3}{4}} = 125$

Step 8

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{3}{4}})}^{\frac{4}{3}} = 125^{\frac{4}{3}}$

Step 9

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

Multiply the exponents in ${(x^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 125^{\frac{4}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 125^{\frac{4}{3}}$

Rewrite the expression.

$x^{\frac{1}{4} \cdot 4} = 125^{\frac{4}{3}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 125^{\frac{4}{3}}$

Rewrite the expression.

$x^{1} = 125^{\frac{4}{3}}$

Simplify.

$x = 125^{\frac{4}{3}}$

Simplify the right side.

Simplify $125^{\frac{4}{3}}$ .

Simplify the expression.

Rewrite $125$ as $5^{3}$ .

$x = {(5^{3})}^{\frac{4}{3}}$

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

$x = 5^{3 (\frac{4}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x = 5^{3 (\frac{4}{3})}$

Rewrite the expression.

$x = 5^{4}$

Raise $5$ to the power of $4$ .

$x = 625$

Solve by Factoring x^(3/4)=125

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