Revise: Exponents and Radicals

a⁰ = 1, and a⁻ⁿ = 1/aⁿ.

5⁻² = 1/25.

aᵐ · aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ.

x⁵ · x³ = x⁸.

a^(m/n) = (ⁿ√a)ᵐ — denominator is the root, numerator is the power.

8^(2/3) = (∛8)² = 4.

Factor out the largest perfect-power factor: ⁿ√(ab) = ⁿ√a · ⁿ√b.

√50 = 5√2.

Match the bases, then set the exponents equal.

2^(x+1) = 32 = 2⁵ → x = 4.

aⁿ = a used as a factor n times.

3⁴ = 81.

a⁰ = 1; a⁻ⁿ = 1/aⁿ — a negative exponent means reciprocal, not negative value.

2⁻³ = 1/8.

Same base: multiplying adds exponents, dividing subtracts them.

b¹²/b⁵ = b⁷.

(aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ.

(x³)⁴ = x¹².

a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ.

27^(2/3) = 9.

ⁿ√a asks "what number to the nth power gives a?"

∛64 = 4.

ⁿ√(aᵐ) = a^(m/n) — index becomes the denominator.

⁵√(x³) = x^(3/5).

Pull out the largest perfect-power factor of the radicand.

√75 = 5√3.

Only like radicals (same index, same radicand) combine, by adding coefficients.

√8 + √18 = 2√2 + 3√2 = 5√2.

ⁿ√a · ⁿ√b = ⁿ√(ab), then simplify.

√6 · √15 = √90 = 3√10.

Rationalize by multiplying top and bottom by the denominator's radical.

5/√3 = 5√3/3.

If aˣ = aʸ (a > 0, a ≠ 1), then x = y.

5^(2x) = 125 → x = 1.5.

A = A₀(1+r)ᵗ (growth) or A₀(1−r)ᵗ (decay).

$18,000 car, 10%/yr depreciation, 2 yrs → $14,580.