Basic Math Examples

y^{100} y^{4} - 64^{2}

$y^{100} y^{4} - 64^{2}$

Step 1

Rewrite

y^{100} y^{4}

$y^{100} y^{4}$ as

{(y^{50} y^{2})}^{2}

${(y^{50} y^{2})}^{2}$ .

{(y^{50} y^{2})}^{2} - 64^{2}

${(y^{50} y^{2})}^{2} - 64^{2}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = y^{50} y^{2}

$a = y^{50} y^{2}$ and

b = 64

$b = 64$ .

(y^{50} y^{2} + 64) (y^{50} y^{2} - 64)

$(y^{50} y^{2} + 64) (y^{50} y^{2} - 64)$

Step 3

Simplify.

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Step 3.1

Multiply

y^{50}

$y^{50}$ by

y^{2}

$y^{2}$ by adding the exponents.

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Step 3.1.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

(y^{50 + 2} + 64) (y^{50} y^{2} - 64)

$(y^{50 + 2} + 64) (y^{50} y^{2} - 64)$

Step 3.1.2

Add

50

$50$ and

2

$2$ .

(y^{52} + 64) (y^{50} y^{2} - 64)

$(y^{52} + 64) (y^{50} y^{2} - 64)$

(y^{52} + 64) (y^{50} y^{2} - 64)

$(y^{52} + 64) (y^{50} y^{2} - 64)$

Step 3.2

Rewrite

y^{50} y^{2}

$y^{50} y^{2}$ as

{(y^{25} y)}^{2}

${(y^{25} y)}^{2}$ .

(y^{52} + 64) ({(y^{25} y)}^{2} - 64)

$(y^{52} + 64) ({(y^{25} y)}^{2} - 64)$

Step 3.3

Rewrite

64

$64$ as

8^{2}

$8^{2}$ .

(y^{52} + 64) ({(y^{25} y)}^{2} - 8^{2})

$(y^{52} + 64) ({(y^{25} y)}^{2} - 8^{2})$

Step 3.4

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = y^{25} y

$a = y^{25} y$ and

b = 8

$b = 8$ .

(y^{52} + 64) ((y^{25} y + 8) (y^{25} y - 8))

$(y^{52} + 64) ((y^{25} y + 8) (y^{25} y - 8))$

Step 3.5

Factor.

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Step 3.5.1

Simplify.

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Step 3.5.1.1

Multiply

y^{25}

$y^{25}$ by

y

$y$ by adding the exponents.

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Step 3.5.1.1.1

Multiply

y^{25}

$y^{25}$ by

y

$y$ .

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Step 3.5.1.1.1.1

Raise

y

$y$ to the power of

1

$1$ .

(y^{52} + 64) ((y^{25} y^{1} + 8) (y^{25} y - 8))

$(y^{52} + 64) ((y^{25} y^{1} + 8) (y^{25} y - 8))$

Step 3.5.1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(y^{52} + 64) ((y^{25 + 1} + 8) (y^{25} y - 8))$

Step 3.5.1.1.2

Add

$25$ and

$1$ .

$(y^{52} + 64) ((y^{26} + 8) (y^{25} y - 8))$

Step 3.5.1.2

Multiply

$y^{25}$ by

$y$ by adding the exponents.

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Step 3.5.1.2.1

Multiply

$y^{25}$ by

$y$ .

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Step 3.5.1.2.1.1

Raise

$y$ to the power of

$1$ .

$(y^{52} + 64) ((y^{26} + 8) (y^{25} y^{1} - 8))$

Step 3.5.1.2.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(y^{52} + 64) ((y^{26} + 8) (y^{25 + 1} - 8))$

Step 3.5.1.2.2

Add

$25$ and

$1$ .

$(y^{52} + 64) ((y^{26} + 8) (y^{26} - 8))$

Step 3.5.2

Remove unnecessary parentheses.

$(y^{52} + 64) (y^{26} + 8) (y^{26} - 8)$