Precalculus Examples

$\frac{3 x^{2} {(2 x + 5)}^{\frac{1}{2}} - x^{3} \frac{1}{2} {(2 x + 5)}^{- \frac{1}{2}} \cdot 2}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x^{2}$ out of $3 x^{2} {(2 x + 5)}^{\frac{1}{2}} - x^{3} \frac{1}{2} \cdot {(2 x + 5)}^{- \frac{1}{2}} \cdot 2$ .

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Step 1.1.1

Factor $x^{2}$ out of $3 x^{2} {(2 x + 5)}^{\frac{1}{2}}$ .

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}}) - x^{3} \frac{1}{2} \cdot {(2 x + 5)}^{- \frac{1}{2}} \cdot 2}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.1.2

Factor $x^{2}$ out of $- x^{3} \frac{1}{2} \cdot {(2 x + 5)}^{- \frac{1}{2}} \cdot 2$ .

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}}) + x^{2} (- x \frac{1}{2} \cdot {(2 x + 5)}^{- \frac{1}{2}} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.1.3

Factor $x^{2}$ out of $x^{2} (3 {(2 x + 5)}^{\frac{1}{2}}) + x^{2} (- x \frac{1}{2} \cdot {(2 x + 5)}^{- \frac{1}{2}} \cdot 2)$ .

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} - x \frac{1}{2} \cdot {(2 x + 5)}^{- \frac{1}{2}} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.2

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

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} - \frac{x}{2} \cdot {(2 x + 5)}^{- \frac{1}{2}} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} - \frac{x}{2} \cdot \frac{1}{{(2 x + 5)}^{\frac{1}{2}}} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.4

Multiply $\frac{1}{{(2 x + 5)}^{\frac{1}{2}}}$ by $\frac{x}{2}$ .

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} - \frac{x}{{(2 x + 5)}^{\frac{1}{2}} \cdot 2} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Move the leading negative in $- \frac{x}{{(2 x + 5)}^{\frac{1}{2}} \cdot 2}$ into the numerator.

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} + \frac{- x}{{(2 x + 5)}^{\frac{1}{2}} \cdot 2} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.5.2

Factor $2$ out of ${(2 x + 5)}^{\frac{1}{2}} \cdot 2$ .

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} + \frac{- x}{2 \cdot {(2 x + 5)}^{\frac{1}{2}}} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.5.3

Cancel the common factor.

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} + \frac{- x}{2 \cdot {(2 x + 5)}^{\frac{1}{2}}} \cdot 2)}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.5.4

Rewrite the expression.

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} + \frac{- x}{{(2 x + 5)}^{\frac{1}{2}}})}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.6

Move the negative in front of the fraction.

$\frac{x^{2} (3 {(2 x + 5)}^{\frac{1}{2}} - \frac{x}{{(2 x + 5)}^{\frac{1}{2}}})}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.7

To write $3 {(2 x + 5)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(2 x + 5)}^{\frac{1}{2}}}{{(2 x + 5)}^{\frac{1}{2}}}$ .

$\frac{x^{2} (\frac{3 {(2 x + 5)}^{\frac{1}{2}} {(2 x + 5)}^{\frac{1}{2}}}{{(2 x + 5)}^{\frac{1}{2}}} - \frac{x}{{(2 x + 5)}^{\frac{1}{2}}})}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.8

Combine the numerators over the common denominator.

$\frac{x^{2} \frac{3 {(2 x + 5)}^{\frac{1}{2}} {(2 x + 5)}^{\frac{1}{2}} - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9

Rewrite $\frac{3 {(2 x + 5)}^{\frac{1}{2}} {(2 x + 5)}^{\frac{1}{2}} - x}{{(2 x + 5)}^{\frac{1}{2}}}$ in a factored form.

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Step 1.9.1

Multiply ${(2 x + 5)}^{\frac{1}{2}}$ by ${(2 x + 5)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.9.1.1

Move ${(2 x + 5)}^{\frac{1}{2}}$ .

$\frac{x^{2} \frac{3 ({(2 x + 5)}^{\frac{1}{2}} {(2 x + 5)}^{\frac{1}{2}}) - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.1.2

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

$\frac{x^{2} \frac{3 {(2 x + 5)}^{\frac{1}{2} + \frac{1}{2}} - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.1.3

Combine the numerators over the common denominator.

$\frac{x^{2} \frac{3 {(2 x + 5)}^{\frac{1 + 1}{2}} - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.1.4

Add $1$ and $1$ .

$\frac{x^{2} \frac{3 {(2 x + 5)}^{\frac{2}{2}} - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.1.5

Divide $2$ by $2$ .

$\frac{x^{2} \frac{3 {(2 x + 5)}^{1} - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.2

Simplify $3 {(2 x + 5)}^{1}$ .

$\frac{x^{2} \frac{3 (2 x + 5) - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.3

Apply the distributive property.

$\frac{x^{2} \frac{3 (2 x) + 3 \cdot 5 - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.4

Multiply $2$ by $3$ .

$\frac{x^{2} \frac{6 x + 3 \cdot 5 - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.5

Multiply $3$ by $5$ .

$\frac{x^{2} \frac{6 x + 15 - x}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.6

Subtract $x$ from $6 x$ .

$\frac{x^{2} \frac{5 x + 15}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.7

Factor $5$ out of $5 x + 15$ .

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Step 1.9.7.1

Factor $5$ out of $5 x$ .

$\frac{x^{2} \frac{5 (x) + 15}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.7.2

Factor $5$ out of $15$ .

$\frac{x^{2} \frac{5 x + 5 \cdot 3}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 1.9.7.3

Factor $5$ out of $5 x + 5 \cdot 3$ .

$\frac{x^{2} \frac{5 (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}}{{({(2 x + 5)}^{\frac{1}{2}})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Multiply the exponents in ${({(2 x + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.1.1

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

$\frac{x^{2} \frac{5 (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}}{{(2 x + 5)}^{\frac{1}{2} \cdot 2}}$

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Cancel the common factor.

$\frac{x^{2} \frac{5 (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}}{{(2 x + 5)}^{\frac{1}{2} \cdot 2}}$

Step 2.1.2.2

Rewrite the expression.

$\frac{x^{2} \frac{5 (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}}{{(2 x + 5)}^{1}}$

Step 2.2

Simplify.

$\frac{x^{2} \frac{5 (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}}{2 x + 5}$

Step 3

Combine $x^{2}$ and $\frac{5 (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}$ .

$\frac{\frac{x^{2} (5 (x + 3))}{{(2 x + 5)}^{\frac{1}{2}}}}{2 x + 5}$

Step 4

Move $5$ to the left of $x^{2}$ .

$\frac{\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}}{2 x + 5}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}} \cdot \frac{1}{2 x + 5}$

Step 6

Multiply $\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}} \cdot \frac{1}{2 x + 5}$ .

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Step 6.1

Multiply $\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2}}}$ by $\frac{1}{2 x + 5}$ .

$\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2}} (2 x + 5)}$

Step 6.2

Multiply ${(2 x + 5)}^{\frac{1}{2}}$ by $2 x + 5$ by adding the exponents.

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Step 6.2.1

Multiply ${(2 x + 5)}^{\frac{1}{2}}$ by $2 x + 5$ .

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Step 6.2.1.1

Raise $2 x + 5$ to the power of $1$ .

$\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2}} {(2 x + 5)}^{1}}$

Step 6.2.1.2

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

$\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2} + 1}}$

Step 6.2.2

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

$\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 6.2.3

Combine the numerators over the common denominator.

$\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{1 + 2}{2}}}$

Step 6.2.4

Add $1$ and $2$ .

$\frac{5 x^{2} (x + 3)}{{(2 x + 5)}^{\frac{3}{2}}}$