Algebra Examples

${(2 \frac{2}{9} a^{3 m - \frac{2}{3}})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 1

Convert $2 \frac{2}{9}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

${((2 + \frac{2}{9}) a^{3 m - \frac{2}{3}})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 1.2

Add $2$ and $\frac{2}{9}$ .

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Step 1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

${((2 \cdot \frac{9}{9} + \frac{2}{9}) a^{3 m - \frac{2}{3}})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 1.2.2

Combine $2$ and $\frac{9}{9}$ .

${((\frac{2 \cdot 9}{9} + \frac{2}{9}) a^{3 m - \frac{2}{3}})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 1.2.3

Combine the numerators over the common denominator.

${(\frac{2 \cdot 9 + 2}{9} a^{3 m - \frac{2}{3}})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $2$ by $9$ .

${(\frac{18 + 2}{9} a^{3 m - \frac{2}{3}})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 1.2.4.2

Add $18$ and $2$ .

${(\frac{20}{9} a^{3 m - \frac{2}{3}})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 2

Combine $\frac{20}{9}$ and $a^{3 m - \frac{2}{3}}$ .

${(\frac{20 a^{3 m - \frac{2}{3}}}{9})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 3

Simplify the numerator.

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Step 3.1

To write $3 m$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{20 a^{3 m \cdot \frac{3}{3} - \frac{2}{3}}}{9})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 3.2

Combine $3 m$ and $\frac{3}{3}$ .

${(\frac{20 a^{\frac{3 m \cdot 3}{3} - \frac{2}{3}}}{9})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 3.3

Combine the numerators over the common denominator.

${(\frac{20 a^{\frac{3 m \cdot 3 - 2}{3}}}{9})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 3.4

Multiply $3$ by $3$ .

${(\frac{20 a^{\frac{9 m - 2}{3}}}{9})}^{3} \cdot {(0.45 a^{m - 5})}^{3}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{20 a^{\frac{9 m - 2}{3}}}{9}$ .

$\frac{{(20 a^{\frac{9 m - 2}{3}})}^{3}}{9^{3}} \cdot {(0.45 a^{m - 5})}^{3}$

Step 4.2

Apply the product rule to $20 a^{\frac{9 m - 2}{3}}$ .

$\frac{20^{3} {(a^{\frac{9 m - 2}{3}})}^{3}}{9^{3}} \cdot {(0.45 a^{m - 5})}^{3}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $20$ to the power of $3$ .

$\frac{8000 {(a^{\frac{9 m - 2}{3}})}^{3}}{9^{3}} \cdot {(0.45 a^{m - 5})}^{3}$

Step 5.2

Multiply the exponents in ${(a^{\frac{9 m - 2}{3}})}^{3}$ .

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Step 5.2.1

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

$\frac{8000 a^{\frac{9 m - 2}{3} \cdot 3}}{9^{3}} \cdot {(0.45 a^{m - 5})}^{3}$

Step 5.2.2

Cancel the common factor of $3$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{8000 a^{\frac{9 m - 2}{3} \cdot 3}}{9^{3}} \cdot {(0.45 a^{m - 5})}^{3}$

Step 5.2.2.2

Rewrite the expression.

$\frac{8000 a^{9 m - 2}}{9^{3}} \cdot {(0.45 a^{m - 5})}^{3}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Raise $9$ to the power of $3$ .

$\frac{8000 a^{9 m - 2}}{729} \cdot {(0.45 a^{m - 5})}^{3}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Apply the product rule to $0.45 a^{m - 5}$ .

$\frac{8000 a^{9 m - 2}}{729} \cdot ({0.45}^{3} {(a^{m - 5})}^{3})$

Step 6.2.2

Rewrite using the commutative property of multiplication.

${0.45}^{3} \frac{8000 a^{9 m - 2}}{729} {(a^{m - 5})}^{3}$

Step 6.2.3

Raise $0.45$ to the power of $3$ .

$0.091125 \frac{8000 a^{9 m - 2}}{729} {(a^{m - 5})}^{3}$

Step 6.3

Cancel the common factor of $0.091125$ .

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Step 6.3.1

Factor $0.091125$ out of $729$ .

$0.091125 \frac{8000 a^{9 m - 2}}{0.091125 (8000)} {(a^{m - 5})}^{3}$

Step 6.3.2

Cancel the common factor.

$0.091125 \frac{8000 a^{9 m - 2}}{0.091125 \cdot 8000} {(a^{m - 5})}^{3}$

Step 6.3.3

Rewrite the expression.

$\frac{8000 a^{9 m - 2}}{8000} {(a^{m - 5})}^{3}$

Step 6.4

Cancel the common factor of $8000$ .

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Step 6.4.1

Cancel the common factor.

$\frac{8000 a^{9 m - 2}}{8000} {(a^{m - 5})}^{3}$

Step 6.4.2

Divide $a^{9 m - 2}$ by $1$ .

$a^{9 m - 2} {(a^{m - 5})}^{3}$

Step 6.5

Multiply the exponents in ${(a^{m - 5})}^{3}$ .

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Step 6.5.1

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

$a^{9 m - 2} a^{(m - 5) \cdot 3}$

Step 6.5.2

Apply the distributive property.

$a^{9 m - 2} a^{m \cdot 3 - 5 \cdot 3}$

Step 6.5.3

Move $3$ to the left of $m$ .

$a^{9 m - 2} a^{3 \cdot m - 5 \cdot 3}$

Step 6.5.4

Multiply $- 5$ by $3$ .

$a^{9 m - 2} a^{3 m - 15}$

Step 7

Multiply $a^{9 m - 2}$ by $a^{3 m - 15}$ by adding the exponents.

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Step 7.1

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

$a^{9 m - 2 + 3 m - 15}$

Step 7.2

Add $9 m$ and $3 m$ .

$a^{12 m - 2 - 15}$

Step 7.3

Subtract $15$ from $- 2$ .

$a^{12 m - 17}$