Math · 08 · Advanced Math

Exponents

Notes

Same base: multiply → add exponents; divide → subtract; power of a power → multiply.
Negative exponent = reciprocal; $x^0=1$ .
Fractional exponent: denominator is the root, $x^{3/2}=\left(\sqrt{x}\right)^3$ .
Rewrite to a common base: $8^x=2^{3x}$ , then set exponents equal.
Growth/decay $a(1\pm r)^t$ : $a$ = start, $1\pm r$ = factor per period.
Factor $1.06$ = 6% growth; $0.94$ = 6% decay.
Doubles every 3 years → $2^{t/3}$ , with $t$ in years.

Product and quotient

x^a\cdot x^b=x^{a+b}\qquad\dfrac{x^a}{x^b}=x^{a-b}

Power rules

(x^a)^b=x^{ab}\qquad(xy)^a=x^ay^a

Negative exponent

x^{-a}=\dfrac{1}{x^a}

Fractional exponent

x^{m/n}=\sqrt[n]{x^m}=\left(\sqrt[n]{x}\right)^m

Exponential model

f(t)=a(1+r)^t

r > 0 growth, r < 0 decay

Doubling / half-life

A=A_0\cdot2^{t/d}\qquad A=A_0\left(\tfrac{1}{2}\right)^{t/h}

Compound interest

A=P\left(1+\dfrac{r}{n}\right)^{nt}

Not on the reference sheet

Match an exponential model by checking two things: the value at t = 0 and the factor per time unit.

10 official SAT questions on Exponents, easiest first.