Basic Math Examples

${(\frac{2}{3} t - \frac{8}{3})}^{3}$

Step 1

Combine $\frac{2}{3}$ and $t$ .

${(\frac{2 t}{3} - \frac{8}{3})}^{3}$

Step 2

Use the Binomial Theorem.

${(\frac{2 t}{3})}^{3} + 3 {(\frac{2 t}{3})}^{2} (- \frac{8}{3}) + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

Apply the product rule to $\frac{2 t}{3}$ .

$\frac{{(2 t)}^{3}}{3^{3}} + 3 {(\frac{2 t}{3})}^{2} (- \frac{8}{3}) + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.1.2

Apply the product rule to $2 t$ .

$\frac{2^{3} t^{3}}{3^{3}} + 3 {(\frac{2 t}{3})}^{2} (- \frac{8}{3}) + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.2

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

$\frac{8 t^{3}}{3^{3}} + 3 {(\frac{2 t}{3})}^{2} (- \frac{8}{3}) + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.3

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

$\frac{8 t^{3}}{27} + 3 {(\frac{2 t}{3})}^{2} (- \frac{8}{3}) + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$\frac{8 t^{3}}{27} + 3 {(\frac{2 t}{3})}^{2} \frac{- 8}{3} + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.4.2

Factor $3$ out of $3 {(\frac{2 t}{3})}^{2}$ .

$\frac{8 t^{3}}{27} + 3 ({(\frac{2 t}{3})}^{2}) \frac{- 8}{3} + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.4.3

Cancel the common factor.

$\frac{8 t^{3}}{27} + 3 {(\frac{2 t}{3})}^{2} \frac{- 8}{3} + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.4.4

Rewrite the expression.

$\frac{8 t^{3}}{27} + {(\frac{2 t}{3})}^{2} \cdot - 8 + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.5

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

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Step 3.5.1

Apply the product rule to $\frac{2 t}{3}$ .

$\frac{8 t^{3}}{27} + \frac{{(2 t)}^{2}}{3^{2}} \cdot - 8 + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.5.2

Apply the product rule to $2 t$ .

$\frac{8 t^{3}}{27} + \frac{2^{2} t^{2}}{3^{2}} \cdot - 8 + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.6

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

$\frac{8 t^{3}}{27} + \frac{4 t^{2}}{3^{2}} \cdot - 8 + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.7

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

$\frac{8 t^{3}}{27} + \frac{4 t^{2}}{9} \cdot - 8 + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.8

Multiply $\frac{4 t^{2}}{9} \cdot - 8$ .

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Step 3.8.1

Combine $\frac{4 t^{2}}{9}$ and $- 8$ .

$\frac{8 t^{3}}{27} + \frac{4 t^{2} \cdot - 8}{9} + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.8.2

Multiply $- 8$ by $4$ .

$\frac{8 t^{3}}{27} + \frac{- 32 t^{2}}{9} + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.9

Move the negative in front of the fraction.

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.10

Cancel the common factor of $3$ .

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Step 3.10.1

Cancel the common factor.

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 3 \frac{2 t}{3} {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.10.2

Rewrite the expression.

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 2 t {(- \frac{8}{3})}^{2} + {(- \frac{8}{3})}^{3}$

Step 3.11

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

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Step 3.11.1

Apply the product rule to $- \frac{8}{3}$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 2 t ({(- 1)}^{2} {(\frac{8}{3})}^{2}) + {(- \frac{8}{3})}^{3}$

Step 3.11.2

Apply the product rule to $\frac{8}{3}$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 2 t ({(- 1)}^{2} \frac{8^{2}}{3^{2}}) + {(- \frac{8}{3})}^{3}$

Step 3.12

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

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 2 t (1 \frac{8^{2}}{3^{2}}) + {(- \frac{8}{3})}^{3}$

Step 3.13

Multiply $\frac{8^{2}}{3^{2}}$ by $1$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 2 t \frac{8^{2}}{3^{2}} + {(- \frac{8}{3})}^{3}$

Step 3.14

Raise $8$ to the power of $2$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 2 t \frac{64}{3^{2}} + {(- \frac{8}{3})}^{3}$

Step 3.15

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

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + 2 t \frac{64}{9} + {(- \frac{8}{3})}^{3}$

Step 3.16

Multiply $2 t \frac{64}{9}$ .

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Step 3.16.1

Combine $2$ and $\frac{64}{9}$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + t \frac{2 \cdot 64}{9} + {(- \frac{8}{3})}^{3}$

Step 3.16.2

Multiply $2$ by $64$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + t \frac{128}{9} + {(- \frac{8}{3})}^{3}$

Step 3.16.3

Combine $t$ and $\frac{128}{9}$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + \frac{t \cdot 128}{9} + {(- \frac{8}{3})}^{3}$

Step 3.17

Move $128$ to the left of $t$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + \frac{128 \cdot t}{9} + {(- \frac{8}{3})}^{3}$

Step 3.18

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

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Step 3.18.1

Apply the product rule to $- \frac{8}{3}$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + \frac{128 t}{9} + {(- 1)}^{3} {(\frac{8}{3})}^{3}$

Step 3.18.2

Apply the product rule to $\frac{8}{3}$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + \frac{128 t}{9} + {(- 1)}^{3} \frac{8^{3}}{3^{3}}$

Step 3.19

Raise $- 1$ to the power of $3$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + \frac{128 t}{9} - \frac{8^{3}}{3^{3}}$

Step 3.20

Raise $8$ to the power of $3$ .

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + \frac{128 t}{9} - \frac{512}{3^{3}}$

Step 3.21

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

$\frac{8 t^{3}}{27} - \frac{32 t^{2}}{9} + \frac{128 t}{9} - \frac{512}{27}$