Finite Math Examples

$({(- \frac{1}{25})}^{2} \cdot {(\frac{1}{25})}^{3}) \cdot {(\frac{25}{3})}^{5}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $- \frac{1}{25}$ .

${(- 1)}^{2} {(\frac{1}{25})}^{2} \cdot {(\frac{1}{25})}^{3} \cdot {(\frac{25}{3})}^{5}$

Step 1.2

Apply the product rule to $\frac{1}{25}$ .

${(- 1)}^{2} \frac{1^{2}}{25^{2}} \cdot {(\frac{1}{25})}^{3} \cdot {(\frac{25}{3})}^{5}$

Step 2

Simplify the expression.

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Step 2.1

Raise $- 1$ to the power of $2$ .

$1 \frac{1^{2}}{25^{2}} \cdot {(\frac{1}{25})}^{3} \cdot {(\frac{25}{3})}^{5}$

Step 2.2

Multiply $\frac{1^{2}}{25^{2}}$ by $1$ .

$\frac{1^{2}}{25^{2}} \cdot {(\frac{1}{25})}^{3} \cdot {(\frac{25}{3})}^{5}$

Step 2.3

One to any power is one.

$\frac{1}{25^{2}} \cdot {(\frac{1}{25})}^{3} \cdot {(\frac{25}{3})}^{5}$

Step 2.4

Raise $25$ to the power of $2$ .

$\frac{1}{625} \cdot {(\frac{1}{25})}^{3} \cdot {(\frac{25}{3})}^{5}$

Step 2.5

Apply the product rule to $\frac{1}{25}$ .

$\frac{1}{625} \cdot \frac{1^{3}}{25^{3}} \cdot {(\frac{25}{3})}^{5}$

Step 3

Combine.

$\frac{1 \cdot 1^{3}}{625 \cdot 25^{3}} \cdot {(\frac{25}{3})}^{5}$

Step 4

Multiply $1$ by $1^{3}$ by adding the exponents.

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Step 4.1

Multiply $1$ by $1^{3}$ .

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Step 4.1.1

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

$\frac{1^{1} \cdot 1^{3}}{625 \cdot 25^{3}} \cdot {(\frac{25}{3})}^{5}$

Step 4.1.2

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

$\frac{1^{1 + 3}}{625 \cdot 25^{3}} \cdot {(\frac{25}{3})}^{5}$

Step 4.2

Add $1$ and $3$ .

$\frac{1^{4}}{625 \cdot 25^{3}} \cdot {(\frac{25}{3})}^{5}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $625$ as $5^{4}$ .

$\frac{1^{4}}{5^{4} \cdot 25^{3}} \cdot {(\frac{25}{3})}^{5}$

Step 5.2

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

$\frac{1^{4}}{5^{4} \cdot {(5^{2})}^{3}} \cdot {(\frac{25}{3})}^{5}$

Step 5.3

Multiply the exponents in ${(5^{2})}^{3}$ .

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Step 5.3.1

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

$\frac{1^{4}}{5^{4} \cdot 5^{2 \cdot 3}} \cdot {(\frac{25}{3})}^{5}$

Step 5.3.2

Multiply $2$ by $3$ .

$\frac{1^{4}}{5^{4} \cdot 5^{6}} \cdot {(\frac{25}{3})}^{5}$

Step 5.4

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

$\frac{1^{4}}{5^{4 + 6}} \cdot {(\frac{25}{3})}^{5}$

Step 5.5

Add $4$ and $6$ .

$\frac{1^{4}}{5^{10}} \cdot {(\frac{25}{3})}^{5}$

Step 6

Simplify terms.

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Step 6.1

One to any power is one.

$\frac{1}{5^{10}} \cdot {(\frac{25}{3})}^{5}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Raise $5$ to the power of $10$ .

$\frac{1}{9765625} \cdot {(\frac{25}{3})}^{5}$

Step 6.2.2

Apply the product rule to $\frac{25}{3}$ .

$\frac{1}{9765625} \cdot \frac{25^{5}}{3^{5}}$

Step 6.3

Combine.

$\frac{1 \cdot 25^{5}}{9765625 \cdot 3^{5}}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Multiply $25^{5}$ by $1$ .

$\frac{25^{5}}{9765625 \cdot 3^{5}}$

Step 6.4.2

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

$\frac{25^{5}}{9765625 \cdot 243}$

Step 6.4.3

Raise $25$ to the power of $5$ .

$\frac{9765625}{9765625 \cdot 243}$

Step 6.4.4

Multiply $9765625$ by $243$ .

$\frac{9765625}{2373046875}$

Step 6.5

Cancel the common factor of $9765625$ and $2373046875$ .

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Step 6.5.1

Factor $9765625$ out of $9765625$ .

$\frac{9765625 (1)}{2373046875}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Factor $9765625$ out of $2373046875$ .

$\frac{9765625 \cdot 1}{9765625 \cdot 243}$

Step 6.5.2.2

Cancel the common factor.

$\frac{9765625 \cdot 1}{9765625 \cdot 243}$

Step 6.5.2.3

Rewrite the expression.

$\frac{1}{243}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{243}$

Decimal Form:

$0.00411522 \dots$