Precalculus Examples

$\frac{(2 x) {(9 + x)}^{8} - 8 {(9 + x)}^{7} x^{2}}{{({(9 + x)}^{8})}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 x {(9 + x)}^{7}$ out of $(2 x) {(9 + x)}^{8} - 8 {(9 + x)}^{7} x^{2}$ .

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Step 1.1.1

Factor $2 x {(9 + x)}^{7}$ out of $(2 x) {(9 + x)}^{8}$ .

$\frac{2 x {(9 + x)}^{7} (9 + x) - 8 {(9 + x)}^{7} x^{2}}{{({(9 + x)}^{8})}^{2}}$

Step 1.1.2

Factor $2 x {(9 + x)}^{7}$ out of $- 8 {(9 + x)}^{7} x^{2}$ .

$\frac{2 x {(9 + x)}^{7} (9 + x) + 2 x {(9 + x)}^{7} (- 4 x)}{{({(9 + x)}^{8})}^{2}}$

Step 1.1.3

Factor $2 x {(9 + x)}^{7}$ out of $2 x {(9 + x)}^{7} (9 + x) + 2 x {(9 + x)}^{7} (- 4 x)$ .

$\frac{2 x {(9 + x)}^{7} (9 + x - 4 x)}{{({(9 + x)}^{8})}^{2}}$

Step 1.2

Subtract $4 x$ from $x$ .

$\frac{2 x {(9 + x)}^{7} (9 - 3 x)}{{({(9 + x)}^{8})}^{2}}$

Step 1.3

Factor $3$ out of $9 - 3 x$ .

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Step 1.3.1

Factor $3$ out of $9$ .

$\frac{2 x {(9 + x)}^{7} (3 (3) - 3 x)}{{({(9 + x)}^{8})}^{2}}$

Step 1.3.2

Factor $3$ out of $- 3 x$ .

$\frac{2 x {(9 + x)}^{7} (3 (3) + 3 (- x))}{{({(9 + x)}^{8})}^{2}}$

Step 1.3.3

Factor $3$ out of $3 (3) + 3 (- x)$ .

$\frac{2 x {(9 + x)}^{7} (3 (3 - x))}{{({(9 + x)}^{8})}^{2}}$

$\frac{2 x {(9 + x)}^{7} \cdot 3 (3 - x)}{{({(9 + x)}^{8})}^{2}}$

Step 1.4

Multiply $3$ by $2$ .

$\frac{6 x {(9 + x)}^{7} (3 - x)}{{({(9 + x)}^{8})}^{2}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Multiply the exponents in ${({(9 + x)}^{8})}^{2}$ .

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Step 2.1.1

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

$\frac{6 x {(9 + x)}^{7} (3 - x)}{{(9 + x)}^{8 \cdot 2}}$

Step 2.1.2

Multiply $8$ by $2$ .

$\frac{6 x {(9 + x)}^{7} (3 - x)}{{(9 + x)}^{16}}$

Step 2.2

Cancel the common factor of ${(9 + x)}^{7}$ and ${(9 + x)}^{16}$ .

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Step 2.2.1

Factor ${(9 + x)}^{7}$ out of $6 x {(9 + x)}^{7} (3 - x)$ .

$\frac{{(9 + x)}^{7} (6 x (3 - x))}{{(9 + x)}^{16}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor ${(9 + x)}^{7}$ out of ${(9 + x)}^{16}$ .

$\frac{{(9 + x)}^{7} (6 x (3 - x))}{{(9 + x)}^{7} {(9 + x)}^{9}}$

Step 2.2.2.2

Cancel the common factor.

$\frac{{(9 + x)}^{7} (6 x (3 - x))}{{(9 + x)}^{7} {(9 + x)}^{9}}$

Step 2.2.2.3

Rewrite the expression.

$\frac{6 x (3 - x)}{{(9 + x)}^{9}}$