Algebra Examples

$\frac{{(10^{x})}^{\frac{2}{3}}}{10^{\frac{1}{3}}} = 10$

Step 1

Multiply both sides of the equation by $10^{\frac{1}{3}}$ .

$10^{\frac{1}{3}} \frac{{(10^{x})}^{\frac{2}{3}}}{10^{\frac{1}{3}}} = 10^{\frac{1}{3}} \cdot 10$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $10^{\frac{1}{3}} \frac{{(10^{x})}^{\frac{2}{3}}}{10^{\frac{1}{3}}}$ .

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Step 2.1.1.1

Cancel the common factor of $10^{\frac{1}{3}}$ .

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Step 2.1.1.1.1

Cancel the common factor.

$10^{\frac{1}{3}} \frac{{(10^{x})}^{\frac{2}{3}}}{10^{\frac{1}{3}}} = 10^{\frac{1}{3}} \cdot 10$

Step 2.1.1.1.2

Rewrite the expression.

${(10^{x})}^{\frac{2}{3}} = 10^{\frac{1}{3}} \cdot 10$

Step 2.1.1.2

Multiply the exponents in ${(10^{x})}^{\frac{2}{3}}$ .

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Step 2.1.1.2.1

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

$10^{x \frac{2}{3}} = 10^{\frac{1}{3}} \cdot 10$

Step 2.1.1.2.2

Combine $x$ and $\frac{2}{3}$ .

$10^{\frac{x \cdot 2}{3}} = 10^{\frac{1}{3}} \cdot 10$

Step 2.1.1.2.3

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

$10^{\frac{2 x}{3}} = 10^{\frac{1}{3}} \cdot 10$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $10^{\frac{1}{3}}$ by $10$ by adding the exponents.

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Step 2.2.1.1

Multiply $10^{\frac{1}{3}}$ by $10$ .

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Step 2.2.1.1.1

Raise $10$ to the power of $1$ .

$10^{\frac{2 x}{3}} = 10^{\frac{1}{3}} \cdot 10^{1}$

Step 2.2.1.1.2

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

$10^{\frac{2 x}{3}} = 10^{\frac{1}{3} + 1}$

Step 2.2.1.2

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

$10^{\frac{2 x}{3}} = 10^{\frac{1}{3} + \frac{3}{3}}$

Step 2.2.1.3

Combine the numerators over the common denominator.

$10^{\frac{2 x}{3}} = 10^{\frac{1 + 3}{3}}$

Step 2.2.1.4

Add $1$ and $3$ .

$10^{\frac{2 x}{3}} = 10^{\frac{4}{3}}$

Step 3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$\frac{2 x}{3} = \frac{4}{3}$

Step 4

Solve for $x$ .

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Step 4.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x = 4$

Step 4.2

Divide each term in $2 x = 4$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $4$ by $2$ .

$x = 2$