Basic Math Examples

${(2 y^{2} - 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1

Simplify each term.

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Step 1.1

Use the Binomial Theorem.

${(2 y^{2})}^{3} + 3 {(2 y^{2})}^{2} \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $2 y^{2}$ .

$2^{3} {(y^{2})}^{3} + 3 {(2 y^{2})}^{2} \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.2

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

$8 {(y^{2})}^{3} + 3 {(2 y^{2})}^{2} \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.3

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

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Step 1.2.3.1

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

$8 y^{2 \cdot 3} + 3 {(2 y^{2})}^{2} \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.3.2

Multiply $2$ by $3$ .

$8 y^{6} + 3 {(2 y^{2})}^{2} \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.4

Apply the product rule to $2 y^{2}$ .

$8 y^{6} + 3 (2^{2} {(y^{2})}^{2}) \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.5

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

$8 y^{6} + 3 (4 {(y^{2})}^{2}) \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.6

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

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Step 1.2.6.1

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

$8 y^{6} + 3 (4 y^{2 \cdot 2}) \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.6.2

Multiply $2$ by $2$ .

$8 y^{6} + 3 (4 y^{4}) \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.7

Multiply $4$ by $3$ .

$8 y^{6} + 12 y^{4} \cdot - 3 + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.8

Multiply $- 3$ by $12$ .

$8 y^{6} - 36 y^{4} + 3 (2 y^{2}) {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.9

Multiply $2$ by $3$ .

$8 y^{6} - 36 y^{4} + 6 y^{2} {(- 3)}^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.10

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

$8 y^{6} - 36 y^{4} + 6 y^{2} \cdot 9 + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.11

Multiply $9$ by $6$ .

$8 y^{6} - 36 y^{4} + 54 y^{2} + {(- 3)}^{3} + 3 (y^{2} - 1) - 5$

Step 1.2.12

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

$8 y^{6} - 36 y^{4} + 54 y^{2} - 27 + 3 (y^{2} - 1) - 5$

Step 1.3

Apply the distributive property.

$8 y^{6} - 36 y^{4} + 54 y^{2} - 27 + 3 y^{2} + 3 \cdot - 1 - 5$

Step 1.4

Multiply $3$ by $- 1$ .

$8 y^{6} - 36 y^{4} + 54 y^{2} - 27 + 3 y^{2} - 3 - 5$

Step 2

Simplify by adding terms.

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Step 2.1

Add $54 y^{2}$ and $3 y^{2}$ .

$8 y^{6} - 36 y^{4} + 57 y^{2} - 27 - 3 - 5$

Step 2.2

Simplify by subtracting numbers.

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Step 2.2.1

Subtract $3$ from $- 27$ .

$8 y^{6} - 36 y^{4} + 57 y^{2} - 30 - 5$

Step 2.2.2

Subtract $5$ from $- 30$ .

$8 y^{6} - 36 y^{4} + 57 y^{2} - 35$