Precalculus Examples

log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{6} - 1)}^{2}}))

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{6} - 1)}^{2}}))$

Step 1

Rewrite

x^{6}

$x^{6}$ as

{(x^{2})}^{3}

${(x^{2})}^{3}$ .

log ⎛ ⎜ ⎜ ⎝ v ⎛ ⎜ ⎜ ⎝ \frac{x^{9} + 2}{(x^{8} + 6) {({(x^{2})}^{3} - 1)}^{2}} ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {({(x^{2})}^{3} - 1)}^{2}}))$

Step 2

Rewrite

1

$1$ as

1^{3}

$1^{3}$ .

log ⎛ ⎜ ⎜ ⎝ v ⎛ ⎜ ⎜ ⎝ \frac{x^{9} + 2}{(x^{8} + 6) {({(x^{2})}^{3} - 1^{3})}^{2}} ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {({(x^{2})}^{3} - 1^{3})}^{2}}))$

Step 3

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

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

a = x^{2}

$a = x^{2}$ and

b = 1

$b = 1$ .

log ⎛ ⎜ ⎜ ⎝ v ⎛ ⎜ ⎜ ⎝ \frac{x^{9} + 2}{(x^{8} + 6) {((x^{2} - 1) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))}^{2}} ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {((x^{2} - 1) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))}^{2}}))$

Step 4

Simplify.

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Step 4.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

log ⎛ ⎜ ⎜ ⎝ v ⎛ ⎜ ⎜ ⎝ \frac{x^{9} + 2}{(x^{8} + 6) {((x^{2} - 1^{2}) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))}^{2}} ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {((x^{2} - 1^{2}) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))}^{2}}))$

Step 4.2

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

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

$a = x$ and

$b = 1$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {((x + 1) (x - 1) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))}^{2}}))$

Step 4.3

Multiply

$x^{2}$ by

$1$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {((x + 1) (x - 1) ({(x^{2})}^{2} + x^{2} + 1^{2}))}^{2}}))$

Step 5

Apply the product rule to

$(x + 1) (x - 1) ({(x^{2})}^{2} + x^{2} + 1^{2})$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {((x + 1) (x - 1))}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 6

Expand

$(x + 1) (x - 1)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x (x - 1) + 1 (x - 1))}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 6.2

Apply the distributive property.

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x \cdot x + x \cdot - 1 + 1 (x - 1))}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 6.3

Apply the distributive property.

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply

$x$ by

$x$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 7.1.2

Move

$- 1$ to the left of

$x$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 7.1.3

Rewrite

$- 1 x$ as

$- x$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - x + 1 x + 1 \cdot - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 7.1.4

Multiply

$x$ by

$1$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - x + x + 1 \cdot - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 7.1.5

Multiply

$- 1$ by

$1$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - x + x - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 7.2

Add

$- x$ and

$x$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} + 0 - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 7.3

Add

$x^{2}$ and

$0$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - 1)}^{2} {({(x^{2})}^{2} + x^{2} + 1^{2})}^{2}}))$

Step 8

Simplify each term.

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Step 8.1

Multiply the exponents in

${(x^{2})}^{2}$ .

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Step 8.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - 1)}^{2} {(x^{2 \cdot 2} + x^{2} + 1^{2})}^{2}}))$

Step 8.1.2

Multiply

$2$ by

$2$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - 1)}^{2} {(x^{4} + x^{2} + 1^{2})}^{2}}))$

Step 8.2

One to any power is one.

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - 1)}^{2} {(x^{4} + x^{2} + 1)}^{2}}))$

Step 9

Rewrite

$1$ as

$1^{2}$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x^{2} - 1^{2})}^{2} {(x^{4} + x^{2} + 1)}^{2}}))$

Step 10

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

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

$a = x$ and

$b = 1$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {((x + 1) (x - 1))}^{2} {(x^{4} + x^{2} + 1)}^{2}}))$

Step 11

Apply the product rule to

$(x + 1) (x - 1)$ .

$\log (v (\frac{x^{9} + 2}{(x^{8} + 6) {(x + 1)}^{2} {(x - 1)}^{2} {(x^{4} + x^{2} + 1)}^{2}}))$