Precalculus Examples

${(\frac{1}{x} - \sqrt{x})}^{4}$

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(\frac{1}{x} - \sqrt{x})}^{4}$ , $n = 4$ so the coefficients of the expansion will correspond with line $5$ .

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 4 - 6 - 4 - 1$ .

$1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4}$

Substitute the actual values of $a$ $\frac{1}{x}$ and $b$ $- \sqrt{x}$ into the expression.

$1 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{0} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Simplify each term.

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Multiply ${(\frac{1}{x})}^{4}$ by $1$ .

${(\frac{1}{x})}^{4} {(- \sqrt{x})}^{0} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

$\frac{1^{4}}{x^{4}} {(- \sqrt{x})}^{0} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

One to any power is one.

$\frac{1}{x^{4}} {(- \sqrt{x})}^{0} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{4}} ({(- 1)}^{0} {\sqrt{x}}^{0}) + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} \frac{1}{x^{4}} {\sqrt{x}}^{0} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Anything raised to $0$ is $1$ .

$1 \frac{1}{x^{4}} {\sqrt{x}}^{0} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Multiply $\frac{1}{x^{4}}$ by $1$ .

$\frac{1}{x^{4}} {\sqrt{x}}^{0} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Anything raised to $0$ is $1$ .

$\frac{1}{x^{4}} \cdot 1 + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Multiply $\frac{1}{x^{4}}$ by $1$ .

$\frac{1}{x^{4}} + 4 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

$\frac{1}{x^{4}} + 4 \frac{1^{3}}{x^{3}} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

One to any power is one.

$\frac{1}{x^{4}} + 4 \frac{1}{x^{3}} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Combine $4$ and $\frac{1}{x^{3}}$ .

$\frac{1}{x^{4}} + \frac{4}{x^{3}} {(- \sqrt{x})}^{1} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Simplify.

$\frac{1}{x^{4}} + \frac{4}{x^{3}} (- \sqrt{x}) + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite using the commutative property of multiplication.

$\frac{1}{x^{4}} - \frac{4}{x^{3}} \sqrt{x} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Combine $\sqrt{x}$ and $\frac{4}{x^{3}}$ .

$\frac{1}{x^{4}} - \frac{\sqrt{x} \cdot 4}{x^{3}} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Move $4$ to the left of $\sqrt{x}$ .

$\frac{1}{x^{4}} - \frac{4 \cdot \sqrt{x}}{x^{3}} + 6 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + 6 \frac{1^{2}}{x^{2}} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

One to any power is one.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + 6 \frac{1}{x^{2}} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Combine $6$ and $\frac{1}{x^{2}}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} {(- \sqrt{x})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} ({(- 1)}^{2} {\sqrt{x}}^{2}) + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite using the commutative property of multiplication.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + {(- 1)}^{2} \frac{6}{x^{2}} {\sqrt{x}}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

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

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + 1 \frac{6}{x^{2}} {\sqrt{x}}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Multiply $\frac{6}{x^{2}}$ by $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} {\sqrt{x}}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} {(x^{\frac{1}{2}})}^{2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

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

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} x^{\frac{1}{2} \cdot 2} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} x^{\frac{2}{2}} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} x^{\frac{2}{2}} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite the expression.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} x^{1} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Simplify.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x^{2}} x + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Cancel the common factor of $x$ .

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Factor $x$ out of $x^{2}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x \cdot x} x + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Cancel the common factor.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x \cdot x} x + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite the expression.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + 4 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Simplify.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + 4 \frac{1}{x} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Combine $4$ and $\frac{1}{x}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + \frac{4}{x} {(- \sqrt{x})}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + \frac{4}{x} ({(- 1)}^{3} {\sqrt{x}}^{3}) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite using the commutative property of multiplication.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + {(- 1)}^{3} \frac{4}{x} {\sqrt{x}}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

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

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - \frac{4}{x} {\sqrt{x}}^{3} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite ${\sqrt{x}}^{3}$ as ${(x^{3})}^{\frac{1}{2}}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - \frac{4}{x} \sqrt{x^{3}} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Factor out $x^{2}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - \frac{4}{x} \sqrt{x^{2} x} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Pull terms out from under the radical.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - \frac{4}{x} (x \sqrt{x}) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Cancel the common factor of $x$ .

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Move the leading negative in $- \frac{4}{x}$ into the numerator.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + \frac{- 4}{x} (x \sqrt{x}) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Factor $x$ out of $x \sqrt{x}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + \frac{- 4}{x} (x (\sqrt{x})) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Cancel the common factor.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} + \frac{- 4}{x} (x \sqrt{x}) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Rewrite the expression.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Multiply ${(\frac{1}{x})}^{0}$ by $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{4}$

Apply the product rule to $\frac{1}{x}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + \frac{1^{0}}{x^{0}} {(- \sqrt{x})}^{4}$

Anything raised to $0$ is $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + \frac{1}{x^{0}} {(- \sqrt{x})}^{4}$

Anything raised to $0$ is $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + \frac{1}{1} {(- \sqrt{x})}^{4}$

Divide $1$ by $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + 1 {(- \sqrt{x})}^{4}$

Multiply ${(- \sqrt{x})}^{4}$ by $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + {(- \sqrt{x})}^{4}$

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + {(- 1)}^{4} {\sqrt{x}}^{4}$

Raise $- 1$ to the power of $4$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + 1 {\sqrt{x}}^{4}$

Multiply ${\sqrt{x}}^{4}$ by $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + {\sqrt{x}}^{4}$

Rewrite ${\sqrt{x}}^{4}$ as $x^{2}$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + {(x^{\frac{1}{2}})}^{4}$

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

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + x^{\frac{1}{2} \cdot 4}$

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + x^{\frac{4}{2}}$

Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + x^{\frac{2 \cdot 2}{2}}$

Cancel the common factors.

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Factor $2$ out of $2$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + x^{\frac{2 \cdot 2}{2 (1)}}$

Cancel the common factor.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + x^{\frac{2 \cdot 2}{2 \cdot 1}}$

Rewrite the expression.

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + x^{\frac{2}{1}}$

Divide $2$ by $1$ .

$\frac{1}{x^{4}} - \frac{4 \sqrt{x}}{x^{3}} + \frac{6}{x} - 4 \sqrt{x} + x^{2}$