Algebra Examples

$f (x) = x^{\frac{5}{2}} - x^{\frac{9}{2}}$ , $g (x) = x^{\frac{3}{2}}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{\frac{3}{2}})$ by substituting in the value of $g$ into $f$ .

$f (x^{\frac{3}{2}}) = {(x^{\frac{3}{2}})}^{\frac{5}{2}} - {(x^{\frac{3}{2}})}^{\frac{9}{2}}$

Step 3

Simplify each term.

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Step 3.1

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{5}{2}}$ .

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Step 3.1.1

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

$f (x^{\frac{3}{2}}) = x^{\frac{3}{2} \cdot \frac{5}{2}} - {(x^{\frac{3}{2}})}^{\frac{9}{2}}$

Step 3.1.2

Multiply $\frac{3}{2} \cdot \frac{5}{2}$ .

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Step 3.1.2.1

Multiply $\frac{3}{2}$ by $\frac{5}{2}$ .

$f (x^{\frac{3}{2}}) = x^{\frac{3 \cdot 5}{2 \cdot 2}} - {(x^{\frac{3}{2}})}^{\frac{9}{2}}$

Step 3.1.2.2

Multiply $3$ by $5$ .

$f (x^{\frac{3}{2}}) = x^{\frac{15}{2 \cdot 2}} - {(x^{\frac{3}{2}})}^{\frac{9}{2}}$

Step 3.1.2.3

Multiply $2$ by $2$ .

$f (x^{\frac{3}{2}}) = x^{\frac{15}{4}} - {(x^{\frac{3}{2}})}^{\frac{9}{2}}$

Step 3.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{9}{2}}$ .

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Step 3.2.1

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

$f (x^{\frac{3}{2}}) = x^{\frac{15}{4}} - x^{\frac{3}{2} \cdot \frac{9}{2}}$

Step 3.2.2

Multiply $\frac{3}{2} \cdot \frac{9}{2}$ .

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Step 3.2.2.1

Multiply $\frac{3}{2}$ by $\frac{9}{2}$ .

$f (x^{\frac{3}{2}}) = x^{\frac{15}{4}} - x^{\frac{3 \cdot 9}{2 \cdot 2}}$

Step 3.2.2.2

Multiply $3$ by $9$ .

$f (x^{\frac{3}{2}}) = x^{\frac{15}{4}} - x^{\frac{27}{2 \cdot 2}}$

Step 3.2.2.3

Multiply $2$ by $2$ .

$f (x^{\frac{3}{2}}) = x^{\frac{15}{4}} - x^{\frac{27}{4}}$