Algebra Examples

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

Step 1

Simplify the expression.

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Step 1.1

Apply the product rule to $\frac{2}{b^{2}}$ .

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

Step 1.2

Raise $2$ to the power of $3$ .

$\frac{8}{{(b^{2})}^{3}} \cdot {(\frac{b^{3}}{4})}^{2} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 1.3

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

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Step 1.3.1

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

$\frac{8}{b^{2 \cdot 3}} \cdot {(\frac{b^{3}}{4})}^{2} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 1.3.2

Multiply $2$ by $3$ .

$\frac{8}{b^{6}} \cdot {(\frac{b^{3}}{4})}^{2} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 1.4

Apply the product rule to $\frac{b^{3}}{4}$ .

$\frac{8}{b^{6}} \cdot \frac{{(b^{3})}^{2}}{4^{2}} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 2

Combine.

$\frac{8 {(b^{3})}^{2}}{b^{6} \cdot 4^{2}} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 3

Simplify the expression.

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Step 3.1

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

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Step 3.1.1

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

$\frac{8 b^{3 \cdot 2}}{b^{6} \cdot 4^{2}} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 3.1.2

Multiply $3$ by $2$ .

$\frac{8 b^{6}}{b^{6} \cdot 4^{2}} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 3.2

Raise $4$ to the power of $2$ .

$\frac{8 b^{6}}{b^{6} \cdot 16} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 4

Cancel the common factor of $8$ and $16$ .

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Step 4.1

Factor $8$ out of $8 b^{6}$ .

$\frac{8 (b^{6})}{b^{6} \cdot 16} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $8$ out of $b^{6} \cdot 16$ .

$\frac{8 (b^{6})}{8 (b^{6} \cdot 2)} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 4.2.2

Cancel the common factor.

$\frac{8 b^{6}}{8 (b^{6} \cdot 2)} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 4.2.3

Rewrite the expression.

$\frac{b^{6}}{b^{6} \cdot 2} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 5

Cancel the common factor of $b^{6}$ .

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Step 5.1

Cancel the common factor.

$\frac{b^{6}}{b^{6} \cdot 2} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 5.2

Rewrite the expression.

$\frac{1}{2} \cdot {(\frac{1}{b^{2}})}^{2}$

Step 6

Apply the product rule to $\frac{1}{b^{2}}$ .

$\frac{1}{2} \cdot \frac{1^{2}}{{(b^{2})}^{2}}$

Step 7

Combine.

$\frac{1 \cdot 1^{2}}{2 {(b^{2})}^{2}}$

Step 8

Multiply $1$ by $1^{2}$ by adding the exponents.

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Step 8.1

Multiply $1$ by $1^{2}$ .

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Step 8.1.1

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

$\frac{1^{1} \cdot 1^{2}}{2 {(b^{2})}^{2}}$

Step 8.1.2

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

$\frac{1^{1 + 2}}{2 {(b^{2})}^{2}}$

Step 8.2

Add $1$ and $2$ .

$\frac{1^{3}}{2 {(b^{2})}^{2}}$

Step 9

One to any power is one.

$\frac{1}{2 {(b^{2})}^{2}}$

Step 10

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

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Step 10.1

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

$\frac{1}{2 b^{2 \cdot 2}}$

Step 10.2

Multiply $2$ by $2$ .

$\frac{1}{2 b^{4}}$