Algebra Examples

$6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}} = 6^{\frac{x}{y}}$

Step 1

Rewrite the equation as $6^{\frac{x}{y}} = 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}}$ .

$6^{\frac{x}{y}} = 6^{\frac{1}{3}} \cdot 6^{\frac{1}{4}}$

Step 2

Multiply $6^{\frac{1}{3}}$ by $6^{\frac{1}{4}}$ by adding the exponents.

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Step 2.1

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

$6^{\frac{x}{y}} = 6^{\frac{1}{3} + \frac{1}{4}}$

Step 2.2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$6^{\frac{x}{y}} = 6^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4}}$

Step 2.3

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6^{\frac{x}{y}} = 6^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{3}{3}}$

Step 2.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$6^{\frac{x}{y}} = 6^{\frac{4}{3 \cdot 4} + \frac{1}{4} \cdot \frac{3}{3}}$

Step 2.4.2

Multiply $3$ by $4$ .

$6^{\frac{x}{y}} = 6^{\frac{4}{12} + \frac{1}{4} \cdot \frac{3}{3}}$

Step 2.4.3

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$6^{\frac{x}{y}} = 6^{\frac{4}{12} + \frac{3}{4 \cdot 3}}$

Step 2.4.4

Multiply $4$ by $3$ .

$6^{\frac{x}{y}} = 6^{\frac{4}{12} + \frac{3}{12}}$

Step 2.5

Combine the numerators over the common denominator.

$6^{\frac{x}{y}} = 6^{\frac{4 + 3}{12}}$

Step 2.6

Add $4$ and $3$ .

$6^{\frac{x}{y}} = 6^{\frac{7}{12}}$

Step 3

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

$\frac{x}{y} = \frac{7}{12}$

Step 4

Solve for $y$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 12$

Step 4.1.2

Since $y, 12$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 12$ then find LCM for the variable part $y^{1}$ .

Step 4.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 4.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 4.1.5

The prime factors for $12$ are $2 \cdot 2 \cdot 3$ .

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Step 4.1.5.1

$12$ has factors of $2$ and $6$ .

$2 \cdot 6$

Step 4.1.5.2

$6$ has factors of $2$ and $3$ .

$2 \cdot 2 \cdot 3$

Step 4.1.6

Multiply $2 \cdot 2 \cdot 3$ .

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Step 4.1.6.1

Multiply $2$ by $2$ .

$4 \cdot 3$

Step 4.1.6.2

Multiply $4$ by $3$ .

$12$

Step 4.1.7

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 4.1.8

The LCM of $y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y$

Step 4.1.9

The LCM for $y, 12$ is the numeric part $12$ multiplied by the variable part.

$12 y$

Step 4.2

Multiply each term in $\frac{x}{y} = \frac{7}{12}$ by $12 y$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $\frac{x}{y} = \frac{7}{12}$ by $12 y$ .

$\frac{x}{y} (12 y) = \frac{7}{12} (12 y)$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Rewrite using the commutative property of multiplication.

$12 \frac{x}{y} y = \frac{7}{12} (12 y)$

Step 4.2.2.2

Combine $12$ and $\frac{x}{y}$ .

$\frac{12 x}{y} y = \frac{7}{12} (12 y)$

Step 4.2.2.3

Cancel the common factor of $y$ .

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Step 4.2.2.3.1

Cancel the common factor.

$\frac{12 x}{y} y = \frac{7}{12} (12 y)$

Step 4.2.2.3.2

Rewrite the expression.

$12 x = \frac{7}{12} (12 y)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $12$ .

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Step 4.2.3.1.1

Factor $12$ out of $12 y$ .

$12 x = \frac{7}{12} (12 (y))$

Step 4.2.3.1.2

Cancel the common factor.

$12 x = \frac{7}{12} (12 y)$

Step 4.2.3.1.3

Rewrite the expression.

$12 x = 7 y$

Step 4.3

Solve the equation.

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Step 4.3.1

Rewrite the equation as $7 y = 12 x$ .

$7 y = 12 x$

Step 4.3.2

Divide each term in $7 y = 12 x$ by $7$ and simplify.

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Step 4.3.2.1

Divide each term in $7 y = 12 x$ by $7$ .

$\frac{7 y}{7} = \frac{12 x}{7}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $7$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{12 x}{7}$

Step 4.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{12 x}{7}$

Step 5

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{12 x}{7}$