Algebra Examples

$\frac{- 3 x \cdot {(x)}^{\frac{1}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 {(x)}^{\frac{1}{2}}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $x$ by ${(x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.1

Move ${(x)}^{\frac{1}{2}}$ .

$\frac{- 3 ({(x)}^{\frac{1}{2}} x)}{{(1 - 3 x^{2})}^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 {(x)}^{\frac{1}{2}}}$

Step 1.1.2

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

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Step 1.1.2.1

Raise $x$ to the power of $1$ .

$\frac{- 3 ({(x)}^{\frac{1}{2}} x^{1})}{{(1 - 3 x^{2})}^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 {(x)}^{\frac{1}{2}}}$

Step 1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.3

Write $1$ as a fraction with a common denominator.

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

Step 1.1.4

Combine the numerators over the common denominator.

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

Step 1.1.5

Add $1$ and $2$ .

$\frac{- 3 x^{\frac{3}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 {(x)}^{\frac{1}{2}}}$

Step 1.2

Move the negative in front of the fraction.

$- \frac{3 x^{\frac{3}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 2

To write $- \frac{3 x^{\frac{3}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$- \frac{3 x^{\frac{3}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$

Step 3

To write $- \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}}$ .

$- \frac{3 x^{\frac{3}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}} \cdot \frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} \cdot \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}}$

Step 4

Write each expression with a common denominator of $2 {(1 - 3 x^{2})}^{\frac{1}{2}} x^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3 x^{\frac{3}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}}$ by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$- \frac{3 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})}{{(1 - 3 x^{2})}^{\frac{1}{2}} (2 x^{\frac{1}{2}})} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} \cdot \frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}}$

Step 4.2

Multiply $\frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ by $\frac{{(1 - 3 x^{2})}^{\frac{1}{2}}}{{(1 - 3 x^{2})}^{\frac{1}{2}}}$ .

$- \frac{3 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})}{{(1 - 3 x^{2})}^{\frac{1}{2}} (2 x^{\frac{1}{2}})} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}$

Step 4.3

Reorder the factors of ${(1 - 3 x^{2})}^{\frac{1}{2}} (2 x^{\frac{1}{2}})$ .

$- \frac{3 x^{\frac{3}{2}} (2 x^{\frac{1}{2}})}{2 x^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}} - \frac{{(1 - 3 x^{2})}^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}$

Step 5

Combine the numerators over the common denominator.

$\frac{- 3 x^{\frac{3}{2}} (2 x^{\frac{1}{2}}) - {(1 - 3 x^{2})}^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $- 1$ out of $- 3 x^{\frac{3}{2}} (2 x^{\frac{1}{2}}) - {(1 - 3 x^{2})}^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}$ .

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Step 6.1.1

Reorder the expression.

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Step 6.1.1.1

Move $x^{\frac{3}{2}}$ .

$\frac{- 3 \cdot 2 x^{\frac{3}{2}} x^{\frac{1}{2}} - {(1 - 3 x^{2})}^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}{2 x^{\frac{1}{2}} {(1 - 3 x^{2})}^{\frac{1}{2}}}$

Step 6.1.1.2

Reorder $1$ and $- 3 x^{2}$ .

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

Step 6.1.1.3

Reorder $1$ and $- 3 x^{2}$ .

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

Step 6.1.2

Factor $- 1$ out of $- 3 \cdot 2 x^{\frac{3}{2}} x^{\frac{1}{2}}$ .

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

Step 6.1.3

Factor $- 1$ out of $- {(- 3 x^{2} + 1)}^{\frac{1}{2}} {(- 3 x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 6.1.4

Factor $- 1$ out of $- (3 \cdot 2 x^{\frac{3}{2}} x^{\frac{1}{2}}) - ({(- 3 x^{2} + 1)}^{\frac{1}{2}} {(- 3 x^{2} + 1)}^{\frac{1}{2}})$ .

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

Step 6.2

Multiply $x^{\frac{3}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 6.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.2.3

Combine the numerators over the common denominator.

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

Step 6.2.4

Add $1$ and $3$ .

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

Step 6.2.5

Divide $4$ by $2$ .

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

Step 6.3

Multiply $3$ by $2$ .

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

Step 6.4

Multiply ${(- 3 x^{2} + 1)}^{\frac{1}{2}}$ by ${(- 3 x^{2} + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 6.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.4.2

Combine the numerators over the common denominator.

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

Step 6.4.3

Add $1$ and $1$ .

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

Step 6.4.4

Divide $2$ by $2$ .

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

Step 6.5

Simplify ${(- 3 x^{2} + 1)}^{1}$ .

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

Step 6.6

Subtract $3 x^{2}$ from $6 x^{2}$ .

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

Step 7

Move the negative in front of the fraction.

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