Basic Math Examples

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \sqrt{{(\frac{25}{9})}^{- 1}} \div \frac{3}{5} - (1 - \frac{2}{5})$

Step 1

Find the common denominator.

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Step 1.1

Multiply $\sqrt{{(\frac{25}{9})}^{- 1}} \div \frac{3}{5}$ by $\frac{{(\frac{3}{5})}^{4}}{{(\frac{3}{5})}^{4}}$ .

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \sqrt{{(\frac{25}{9})}^{- 1}} \div \frac{3}{5} \cdot \frac{{(\frac{3}{5})}^{4}}{{(\frac{3}{5})}^{4}} - (1 - \frac{2}{5})$

Step 1.2

Multiply $\sqrt{{(\frac{25}{9})}^{- 1}} \div \frac{3}{5}$ by $\frac{{(\frac{3}{5})}^{4}}{{(\frac{3}{5})}^{4}}$ .

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \frac{\sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4}}{\frac{3}{5} {(\frac{3}{5})}^{4}} - (1 - \frac{2}{5})$

Step 1.3

Write $- (1 - \frac{2}{5})$ as a fraction with denominator $1$ .

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \frac{\sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4}}{\frac{3}{5} {(\frac{3}{5})}^{4}} + \frac{- (1 - \frac{2}{5})}{1}$

Step 1.4

Multiply $\frac{- (1 - \frac{2}{5})}{1}$ by $\frac{{(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$ .

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \frac{\sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4}}{\frac{3}{5} {(\frac{3}{5})}^{4}} + \frac{- (1 - \frac{2}{5})}{1} \cdot \frac{{(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 1.5

Multiply $\frac{- (1 - \frac{2}{5})}{1}$ by $\frac{{(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$ .

Step 1.6

Multiply $\frac{3}{5}$ by ${(\frac{3}{5})}^{4}$ by adding the exponents.

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Step 1.6.1

Multiply $\frac{3}{5}$ by ${(\frac{3}{5})}^{4}$ .

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Step 1.6.1.1

Raise $\frac{3}{5}$ to the power of $1$ .

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \frac{\sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4}}{{(\frac{3}{5})}^{1} {(\frac{3}{5})}^{4}} + \frac{- (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 1.6.1.2

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

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \frac{\sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4}}{{(\frac{3}{5})}^{1 + 4}} + \frac{- (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 1.6.2

Add $1$ and $4$ .

${(\frac{3}{5})}^{7} \div {(\frac{3}{5})}^{5} + \frac{\sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4}}{{(\frac{3}{5})}^{5}} + \frac{- (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 2

Combine the numerators over the common denominator.

$\frac{{(\frac{3}{5})}^{7} + \sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3

Simplify each term.

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Step 3.1

Apply the product rule to $\frac{3}{5}$ .

$\frac{\frac{3^{7}}{5^{7}} + \sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.2

Raise $3$ to the power of $7$ .

$\frac{\frac{2187}{5^{7}} + \sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.3

Raise $5$ to the power of $7$ .

$\frac{\frac{2187}{78125} + \sqrt{{(\frac{25}{9})}^{- 1}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.4

Apply the product rule to $\frac{25}{9}$ .

$\frac{\frac{2187}{78125} + \sqrt{\frac{25^{- 1}}{9^{- 1}}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.5

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

$\frac{\frac{2187}{78125} + \sqrt{\frac{1}{9^{- 1} \cdot 25}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.6

Move $9^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{\frac{2187}{78125} + \sqrt{\frac{9}{25}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.7

Rewrite $\sqrt{\frac{9}{25}}$ as $\frac{\sqrt{9}}{\sqrt{25}}$ .

$\frac{\frac{2187}{78125} + \frac{\sqrt{9}}{\sqrt{25}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{2187}{78125} + \frac{\sqrt{3^{2}}}{\sqrt{25}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.8.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\frac{2187}{78125} + \frac{3}{\sqrt{25}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.9

Simplify the denominator.

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Step 3.9.1

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

$\frac{\frac{2187}{78125} + \frac{3}{\sqrt{5^{2}}} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\frac{2187}{78125} + \frac{3}{5} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.10

Multiply $\frac{3}{5}$ by ${(\frac{3}{5})}^{4}$ by adding the exponents.

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Step 3.10.1

Multiply $\frac{3}{5}$ by ${(\frac{3}{5})}^{4}$ .

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Step 3.10.1.1

Raise $\frac{3}{5}$ to the power of $1$ .

$\frac{\frac{2187}{78125} + {(\frac{3}{5})}^{1} {(\frac{3}{5})}^{4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.10.1.2

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

$\frac{\frac{2187}{78125} + {(\frac{3}{5})}^{1 + 4} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.10.2

Add $1$ and $4$ .

$\frac{\frac{2187}{78125} + {(\frac{3}{5})}^{5} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.11

Apply the product rule to $\frac{3}{5}$ .

$\frac{\frac{2187}{78125} + \frac{3^{5}}{5^{5}} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.12

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

$\frac{\frac{2187}{78125} + \frac{243}{5^{5}} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.13

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

$\frac{\frac{2187}{78125} + \frac{243}{3125} - (1 - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.14

Write $1$ as a fraction with a common denominator.

$\frac{\frac{2187}{78125} + \frac{243}{3125} - (\frac{5}{5} - \frac{2}{5}) {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.15

Combine the numerators over the common denominator.

$\frac{\frac{2187}{78125} + \frac{243}{3125} - \frac{5 - 2}{5} {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.16

Subtract $2$ from $5$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} - \frac{3}{5} {(\frac{3}{5})}^{5}}{{(\frac{3}{5})}^{5}}$

Step 3.17

Multiply $\frac{3}{5}$ by ${(\frac{3}{5})}^{5}$ by adding the exponents.

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Step 3.17.1

Move ${(\frac{3}{5})}^{5}$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} - ({(\frac{3}{5})}^{5} \frac{3}{5})}{{(\frac{3}{5})}^{5}}$

Step 3.17.2

Multiply ${(\frac{3}{5})}^{5}$ by $\frac{3}{5}$ .

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Step 3.17.2.1

Raise $\frac{3}{5}$ to the power of $1$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} - ({(\frac{3}{5})}^{5} {(\frac{3}{5})}^{1})}{{(\frac{3}{5})}^{5}}$

Step 3.17.2.2

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

$\frac{\frac{2187}{78125} + \frac{243}{3125} - {(\frac{3}{5})}^{5 + 1}}{{(\frac{3}{5})}^{5}}$

Step 3.17.3

Add $5$ and $1$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} - {(\frac{3}{5})}^{6}}{{(\frac{3}{5})}^{5}}$

Step 3.18

Apply the product rule to $\frac{3}{5}$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} - \frac{3^{6}}{5^{6}}}{{(\frac{3}{5})}^{5}}$

Step 3.19

Raise $3$ to the power of $6$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} - \frac{729}{5^{6}}}{{(\frac{3}{5})}^{5}}$

Step 3.20

Raise $5$ to the power of $6$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} - \frac{729}{15625}}{{(\frac{3}{5})}^{5}}$

Step 4

Find the common denominator.

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Step 4.1

Multiply $\frac{243}{3125}$ by $\frac{25}{25}$ .

$\frac{\frac{2187}{78125} + \frac{243}{3125} \cdot \frac{25}{25} - \frac{729}{15625}}{{(\frac{3}{5})}^{5}}$

Step 4.2

Multiply $\frac{243}{3125}$ by $\frac{25}{25}$ .

$\frac{\frac{2187}{78125} + \frac{243 \cdot 25}{3125 \cdot 25} - \frac{729}{15625}}{{(\frac{3}{5})}^{5}}$

Step 4.3

Multiply $\frac{729}{15625}$ by $\frac{5}{5}$ .

$\frac{\frac{2187}{78125} + \frac{243 \cdot 25}{3125 \cdot 25} - (\frac{729}{15625} \cdot \frac{5}{5})}{{(\frac{3}{5})}^{5}}$

Step 4.4

Multiply $\frac{729}{15625}$ by $\frac{5}{5}$ .

$\frac{\frac{2187}{78125} + \frac{243 \cdot 25}{3125 \cdot 25} - \frac{729 \cdot 5}{15625 \cdot 5}}{{(\frac{3}{5})}^{5}}$

Step 4.5

Reorder the factors of $3125 \cdot 25$ .

$\frac{\frac{2187}{78125} + \frac{243 \cdot 25}{25 \cdot 3125} - \frac{729 \cdot 5}{15625 \cdot 5}}{{(\frac{3}{5})}^{5}}$

Step 4.6

Multiply $25$ by $3125$ .

$\frac{\frac{2187}{78125} + \frac{243 \cdot 25}{78125} - \frac{729 \cdot 5}{15625 \cdot 5}}{{(\frac{3}{5})}^{5}}$

Step 4.7

Reorder the factors of $15625 \cdot 5$ .

$\frac{\frac{2187}{78125} + \frac{243 \cdot 25}{78125} - \frac{729 \cdot 5}{5 \cdot 15625}}{{(\frac{3}{5})}^{5}}$

Step 4.8

Multiply $5$ by $15625$ .

$\frac{\frac{2187}{78125} + \frac{243 \cdot 25}{78125} - \frac{729 \cdot 5}{78125}}{{(\frac{3}{5})}^{5}}$

Step 5

Combine the numerators over the common denominator.

$\frac{\frac{2187 + 243 \cdot 25 - 729 \cdot 5}{78125}}{{(\frac{3}{5})}^{5}}$

Step 6

Simplify each term.

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Step 6.1

Multiply $243$ by $25$ .

$\frac{\frac{2187 + 6075 - 729 \cdot 5}{78125}}{{(\frac{3}{5})}^{5}}$

Step 6.2

Multiply $- 729$ by $5$ .

$\frac{\frac{2187 + 6075 - 3645}{78125}}{{(\frac{3}{5})}^{5}}$

Step 7

Simplify by adding and subtracting.

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Step 7.1

Add $2187$ and $6075$ .

$\frac{\frac{8262 - 3645}{78125}}{{(\frac{3}{5})}^{5}}$

Step 7.2

Subtract $3645$ from $8262$ .

$\frac{\frac{4617}{78125}}{{(\frac{3}{5})}^{5}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{4617}{78125} \cdot \frac{1}{{(\frac{3}{5})}^{5}}$

Step 9

Combine fractions.

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Step 9.1

Combine.

$\frac{4617 \cdot 1}{78125 {(\frac{3}{5})}^{5}}$

Step 9.2

Multiply $4617$ by $1$ .

$\frac{4617}{78125 {(\frac{3}{5})}^{5}}$

Step 10

Simplify the denominator.

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Step 10.1

Apply the product rule to $\frac{3}{5}$ .

$\frac{4617}{78125 \frac{3^{5}}{5^{5}}}$

Step 10.2

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

$\frac{4617}{78125 \frac{243}{5^{5}}}$

Step 10.3

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

$\frac{4617}{78125 (\frac{243}{3125})}$

Step 11

Simplify terms.

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Step 11.1

Combine $78125$ and $\frac{243}{3125}$ .

$\frac{4617}{\frac{78125 \cdot 243}{3125}}$

Step 11.2

Simplify the expression.

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Step 11.2.1

Multiply $78125$ by $243$ .

$\frac{4617}{\frac{18984375}{3125}}$

Step 11.2.2

Divide $18984375$ by $3125$ .

$\frac{4617}{6075}$

Step 11.3

Cancel the common factor of $4617$ and $6075$ .

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Step 11.3.1

Factor $243$ out of $4617$ .

$\frac{243 (19)}{6075}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $243$ out of $6075$ .

$\frac{243 \cdot 19}{243 \cdot 25}$

Step 11.3.2.2

Cancel the common factor.

$\frac{243 \cdot 19}{243 \cdot 25}$

Step 11.3.2.3

Rewrite the expression.

$\frac{19}{25}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{19}{25}$

Decimal Form:

$0.76$