Finite Math Examples

$3^{2 X} \cdot 3^{5 X} = 27$

Step 1

Subtract $27$ from both sides of the equation.

$3^{2 X} \cdot 3^{5 X} - 27 = 0$

Step 2

Multiply $3^{2 X}$ by $3^{5 X}$ by adding the exponents.

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Step 2.1

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

$3^{2 X + 5 X} - 27 = 0$

Step 2.2

Add $2 X$ and $5 X$ .

$3^{7 X} - 27 = 0$

Step 3

Add $27$ to both sides of the equation.

$3^{7 X} = 27$

Step 4

Create equivalent expressions in the equation that all have equal bases.

$3^{7 X} = 3^{3}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$7 X = 3$

Step 6

Divide each term in $7 X = 3$ by $7$ and simplify.

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Step 6.1

Divide each term in $7 X = 3$ by $7$ .

$\frac{7 X}{7} = \frac{3}{7}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $7$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{7 X}{7} = \frac{3}{7}$

Step 6.2.1.2

Divide $X$ by $1$ .

$X = \frac{3}{7}$