Algebra Examples

$\frac{{(5^{a})}^{2}}{5^{3}} = 125$

Step 1

Multiply both sides of the equation by $5^{3}$ .

$5^{3} \frac{{(5^{a})}^{2}}{5^{3}} = 5^{3} \cdot 125$

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 $5^{3} \frac{{(5^{a})}^{2}}{5^{3}}$ .

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Step 2.1.1.1

Cancel the common factor of $5^{3}$ .

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Step 2.1.1.1.1

Cancel the common factor.

$5^{3} \frac{{(5^{a})}^{2}}{5^{3}} = 5^{3} \cdot 125$

Step 2.1.1.1.2

Rewrite the expression.

${(5^{a})}^{2} = 5^{3} \cdot 125$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

$5^{a \cdot 2} = 5^{3} \cdot 125$

Step 2.1.1.2.2

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

$5^{2 a} = 5^{3} \cdot 125$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $5^{3} \cdot 125$ .

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Step 2.2.1.1

Raise $5$ to the power of $3$ .

$5^{2 a} = 125 \cdot 125$

Step 2.2.1.2

Multiply $125$ by $125$ .

$5^{2 a} = 15625$

Step 3

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

$5^{2 a} = 5^{6}$

Step 4

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

$2 a = 6$

Step 5

Divide each term in $2 a = 6$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 a = 6$ by $2$ .

$\frac{2 a}{2} = \frac{6}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{6}{2}$

Step 5.2.1.2

Divide $a$ by $1$ .

$a = \frac{6}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $6$ by $2$ .

$a = 3$