Basic Math Examples

$\frac{125^{3 k}}{25} = 25$

Step 1

Raise $25$ to the power of $1$ .

$\frac{125^{3 k}}{25^{1}} = 25$

Step 2

Move $25^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$125^{3 k} \cdot 25^{- 1} = 25$

Step 3

Rewrite $125$ as $5^{3}$ .

${(5^{3})}^{3 k} \cdot 25^{- 1} = 25$

Step 4

Multiply the exponents in ${(5^{3})}^{3 k}$ .

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Step 4.1

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

$5^{3 (3 k)} \cdot 25^{- 1} = 25$

Step 4.2

Multiply $3$ by $3$ .

$5^{9 k} \cdot 25^{- 1} = 25$

Step 5

Rewrite $25$ as $5^{2}$ .

$5^{9 k} \cdot {(5^{2})}^{- 1} = 25$

Step 6

Multiply the exponents in ${(5^{2})}^{- 1}$ .

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Step 6.1

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

$5^{9 k} \cdot 5^{2 \cdot - 1} = 25$

Step 6.2

Multiply $2$ by $- 1$ .

$5^{9 k} \cdot 5^{- 2} = 25$

Step 7

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

$5^{9 k - 2} = 25$

Step 8

Create equivalent expressions in the equation that all have equal bases.

$5^{9 k - 2} = 5^{2}$

Step 9

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

$9 k - 2 = 2$

Step 10

Solve for $k$ .

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Step 10.1

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

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Step 10.1.1

Add $2$ to both sides of the equation.

$9 k = 2 + 2$

Step 10.1.2

Add $2$ and $2$ .

$9 k = 4$

Step 10.2

Divide each term in $9 k = 4$ by $9$ and simplify.

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Step 10.2.1

Divide each term in $9 k = 4$ by $9$ .

$\frac{9 k}{9} = \frac{4}{9}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Cancel the common factor of $9$ .

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Step 10.2.2.1.1

Cancel the common factor.

$\frac{9 k}{9} = \frac{4}{9}$

Step 10.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{4}{9}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$k = \frac{4}{9}$

Decimal Form:

$k = 0.\overline{4}$