Algebra Examples

$\sqrt[5]{2 x - 3} - \sqrt[5]{x - 4} = 0$

Step 1

Add $\sqrt[5]{x - 4}$ to both sides of the equation.

$\sqrt[5]{2 x - 3} = \sqrt[5]{x - 4}$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{2 x - 3}}^{5} = {\sqrt[5]{x - 4}}^{5}$

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 $\sqrt[5]{2 x - 3}$ as ${(2 x - 3)}^{\frac{1}{5}}$ .

${({(2 x - 3)}^{\frac{1}{5}})}^{5} = {\sqrt[5]{x - 4}}^{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(2 x - 3)}^{\frac{1}{5}})}^{5}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(2 x - 3)}^{\frac{1}{5}})}^{5}$ .

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Step 3.2.1.1.1

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

${(2 x - 3)}^{\frac{1}{5} \cdot 5} = {\sqrt[5]{x - 4}}^{5}$

Step 3.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(2 x - 3)}^{\frac{1}{5} \cdot 5} = {\sqrt[5]{x - 4}}^{5}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x - 3)}^{1} = {\sqrt[5]{x - 4}}^{5}$

Step 3.2.1.2

Simplify.

$2 x - 3 = {\sqrt[5]{x - 4}}^{5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Rewrite ${\sqrt[5]{x - 4}}^{5}$ as $x - 4$ .

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Step 3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x - 4}$ as ${(x - 4)}^{\frac{1}{5}}$ .

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

Step 3.3.1.2

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

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

Step 3.3.1.3

Combine $\frac{1}{5}$ and $5$ .

$2 x - 3 = {(x - 4)}^{\frac{5}{5}}$

Step 3.3.1.4

Cancel the common factor of $5$ .

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Step 3.3.1.4.1

Cancel the common factor.

$2 x - 3 = {(x - 4)}^{\frac{5}{5}}$

Step 3.3.1.4.2

Rewrite the expression.

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

Step 3.3.1.5

Simplify.

$2 x - 3 = x - 4$

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

Subtract $x$ from both sides of the equation.

$2 x - 3 - x = - 4$

Step 4.1.2

Subtract $x$ from $2 x$ .

$x - 3 = - 4$

Step 4.2

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

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Step 4.2.1

Add $3$ to both sides of the equation.

$x = - 4 + 3$

Step 4.2.2

Add $- 4$ and $3$ .

$x = - 1$