If 3n + 4 − 3n + 2 = 8, what is the value of n?

Concept

When every term shares the same base, use the exponent addition law am+n = am · an. To solve an equation whose terms all have this common base, factor out the smallest power of that base, simplify the remaining numeric coefficient, then express both sides as the same base (using a0 = 1 when the simplified value is 1) and equate the exponents.

Application

1. Rewrite 3n + 4 using the exponent addition law: 3n + 4 = 3n + 2 · 32 = 9 · 3n + 2.

2. Substitute this into the equation: 9 · 3n + 2 − 3n + 2 = 8.

3. Factor out the common term 3n + 2: 3n + 2 · (9 − 1) = 8, i.e. 3n + 2 · 8 = 8.

4. Divide both sides by 8: 3n + 2 = 1.

5. Express 1 as a power of the same base: 1 = 30, so 3n + 2 = 30.

6. Since the bases are equal, equate the exponents: n + 2 = 0.

7. Solve for n: n = −2.

Cross-check

Substitute n = −2 back into the original expression: 3−2 + 4 − 3−2 + 2 = 32 − 30 = 9 − 1 = 8, which matches the right-hand side, confirming the result.