Finite Math Examples

$\frac{3^{a + b} \cdot 3^{a}}{3^{b} + 9^{b}}$

Step 1

Multiply $3^{a + b}$ by $3^{a}$ by adding the exponents.

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Step 1.1

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

$\frac{3^{a + b + a}}{3^{b} + 9^{b}}$

Step 1.2

Add $a$ and $a$ .

$\frac{3^{2 a + b}}{3^{b} + 9^{b}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $3^{b}$ as ${(3^{2})}^{b}$ .

$\frac{3^{2 a + b}}{3^{b} + {(3^{2})}^{b}}$

Step 2.2

Rewrite ${(3^{2})}^{b}$ as ${(3^{b})}^{2}$ .

$\frac{3^{2 a + b}}{3^{b} + {(3^{b})}^{2}}$

Step 2.3

Let $u = 3^{b}$ . Substitute $u$ for all occurrences of $3^{b}$ .

$\frac{3^{2 a + b}}{u + u^{2}}$

Step 2.4

Factor $u$ out of $u + u^{2}$ .

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Step 2.4.1

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

$\frac{3^{2 a + b}}{u^{1} + u^{2}}$

Step 2.4.2

Factor $u$ out of $u^{1}$ .

$\frac{3^{2 a + b}}{u \cdot 1 + u^{2}}$

Step 2.4.3

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

$\frac{3^{2 a + b}}{u \cdot 1 + u \cdot u}$

Step 2.4.4

Factor $u$ out of $u \cdot 1 + u \cdot u$ .

$\frac{3^{2 a + b}}{u (1 + u)}$

Step 2.5

Replace all occurrences of $u$ with $3^{b}$ .

$\frac{3^{2 a + b}}{3^{b} (1 + 3^{b})}$

Step 3

Cancel the common factor of $3^{2 a + b}$ and $3^{b}$ .

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Step 3.1

Factor $3^{b}$ out of $3^{2 a + b}$ .

$\frac{3^{b} \cdot 3^{2 a}}{3^{b} (1 + 3^{b})}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Cancel the common factor.

$\frac{3^{b} \cdot 3^{2 a}}{3^{b} (1 + 3^{b})}$

Step 3.2.2

Rewrite the expression.

$\frac{3^{2 a}}{1 + 3^{b}}$