Finite Math Examples

$\frac{6 k}{{(k^{2})}^{3}} = 2 k^{3}$

Step 1

Factor each term.

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Step 1.1

Multiply the exponents in ${(k^{2})}^{3}$ .

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Step 1.1.1

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

$\frac{6 k}{k^{2 \cdot 3}} = 2 k^{3}$

Step 1.1.2

Multiply $2$ by $3$ .

$\frac{6 k}{k^{6}} = 2 k^{3}$

Step 1.2

Cancel the common factor of $k$ and $k^{6}$ .

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Step 1.2.1

Factor $k$ out of $6 k$ .

$\frac{k \cdot 6}{k^{6}} = 2 k^{3}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $k$ out of $k^{6}$ .

$\frac{k \cdot 6}{k \cdot k^{5}} = 2 k^{3}$

Step 1.2.2.2

Cancel the common factor.

$\frac{k \cdot 6}{k \cdot k^{5}} = 2 k^{3}$

Step 1.2.2.3

Rewrite the expression.

$\frac{6}{k^{5}} = 2 k^{3}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$k^{5}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$k^{5}$

Step 3

Multiply each term in $\frac{6}{k^{5}} = 2 k^{3}$ by $k^{5}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{6}{k^{5}} = 2 k^{3}$ by $k^{5}$ .

$\frac{6}{k^{5}} k^{5} = 2 k^{3} k^{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $k^{5}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6}{k^{5}} k^{5} = 2 k^{3} k^{5}$

Step 3.2.1.2

Rewrite the expression.

$6 = 2 k^{3} k^{5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $k^{3}$ by $k^{5}$ by adding the exponents.

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Step 3.3.1.1

Move $k^{5}$ .

$6 = 2 (k^{5} k^{3})$

Step 3.3.1.2

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

$6 = 2 k^{5 + 3}$

Step 3.3.1.3

Add $5$ and $3$ .

$6 = 2 k^{8}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $2 k^{8} = 6$ .

$2 k^{8} = 6$

Step 4.2

Divide each term in $2 k^{8} = 6$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 k^{8} = 6$ by $2$ .

$\frac{2 k^{8}}{2} = \frac{6}{2}$

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.

$\frac{2 k^{8}}{2} = \frac{6}{2}$

Step 4.2.2.1.2

Divide $k^{8}$ by $1$ .

$k^{8} = \frac{6}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $6$ by $2$ .

$k^{8} = 3$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$k = \pm \sqrt[8]{3}$

Step 4.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.4.1

First, use the positive value of the $\pm$ to find the first solution.

$k = \sqrt[8]{3}$

Step 4.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$k = - \sqrt[8]{3}$

Step 4.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$k = \sqrt[8]{3}, - \sqrt[8]{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$k = \sqrt[8]{3}, - \sqrt[8]{3}$

Decimal Form:

$k = 1.14720269 \dots, - 1.14720269 \dots$