Pre-Algebra Examples

\frac{2^{\frac{1}{3}}}{1^{\frac{1}{2}}} = \frac{x}{2^{\frac{1}{4}}}

$\frac{2^{\frac{1}{3}}}{1^{\frac{1}{2}}} = \frac{x}{2^{\frac{1}{4}}}$

Step 1

Solve for

y

$y$ .

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Step 1.1

Multiply both sides by

2^{\frac{1}{4}}

$2^{\frac{1}{4}}$ .

2^{\frac{1}{3}} \cdot 2^{\frac{1}{4}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{1}{3}} \cdot 2^{\frac{1}{4}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Multiply

2^{\frac{1}{3}}

$2^{\frac{1}{3}}$ by

2^{\frac{1}{4}}

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

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Step 1.2.1.1.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

2^{\frac{1}{3} + \frac{1}{4}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{1}{3} + \frac{1}{4}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.2

To write

\frac{1}{3}

$\frac{1}{3}$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

2^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.3

To write

\frac{1}{4}

$\frac{1}{4}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

2^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{3}{3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{3}{3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.4

Write each expression with a common denominator of

12

$12$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 1.2.1.1.4.1

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

\frac{4}{4}

$\frac{4}{4}$ .

2^{\frac{4}{3 \cdot 4} + \frac{1}{4} \cdot \frac{3}{3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{4}{3 \cdot 4} + \frac{1}{4} \cdot \frac{3}{3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.4.2

Multiply

3

$3$ by

4

$4$ .

2^{\frac{4}{12} + \frac{1}{4} \cdot \frac{3}{3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{4}{12} + \frac{1}{4} \cdot \frac{3}{3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.4.3

Multiply

\frac{1}{4}

$\frac{1}{4}$ by

\frac{3}{3}

$\frac{3}{3}$ .

2^{\frac{4}{12} + \frac{3}{4 \cdot 3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{4}{12} + \frac{3}{4 \cdot 3}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.4.4

Multiply

4

$4$ by

3

$3$ .

2^{\frac{4}{12} + \frac{3}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{4}{12} + \frac{3}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

2^{\frac{4}{12} + \frac{3}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{4}{12} + \frac{3}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.5

Combine the numerators over the common denominator.

2^{\frac{4 + 3}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{4 + 3}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.1.1.6

Add

4

$4$ and

3

$3$ .

2^{\frac{7}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{7}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

2^{\frac{7}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{7}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

2^{\frac{7}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}

$2^{\frac{7}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Cancel the common factor of

2^{\frac{1}{4}}

$2^{\frac{1}{4}}$ .

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Step 1.2.2.1.1

Cancel the common factor.

$2^{\frac{7}{12}} = \frac{x}{2^{\frac{1}{4}}} \cdot 2^{\frac{1}{4}}$

Step 1.2.2.1.2

Rewrite the expression.

$2^{\frac{7}{12}} = x$

Step 1.3

Rewrite the equation as

$x = 2^{\frac{7}{12}}$ .

$x = 2^{\frac{7}{12}}$

Step 2

Since

$x = 2^{\frac{7}{12}}$ is a vertical line, there is no y-intercept and the slope is undefined.

Slope: Undefined

y-intercept: No y-intercept

Step 3

Find two points on the line.

$\begin{array}{c} x & y \\ 2^{\frac{7}{12}} & 0 \\ 2^{\frac{7}{12}} & 1 \end{array}$

Step 4

Graph the line using the slope, y-intercept, and two points.

Slope: Undefined

y-intercept: No y-intercept

$\begin{array}{c} x & y \\ 2^{\frac{7}{12}} & 0 \\ 2^{\frac{7}{12}} & 1 \end{array}$

Step 5