7. Exponents, Radicals & Exponential Models

Exponents are repeated multiplication. The rules the SAT® uses most:

• Multiply like bases: add exponents — \(2^3 \cdot 2^4 = 2^{7} = 128\).
• Divide like bases: subtract exponents — \(\dfrac{x^5}{x^2} = x^{3}\).
• Power of a power: multiply — \((x^3)^2 = x^{6}\).
• Zero power: anything (nonzero) to the 0 is \(1\).
• Negative exponent: \(x^{-2} = \dfrac{1}{x^2}\).

A radical (root) undoes a power: \(\sqrt{64} = 8\) because \(8^2 = 64\). A square root is the same as a \(\tfrac{1}{2}\) power.

Exponential models grow or shrink by a constant factor: \(y = a \cdot b^{x}\). If \(b > 1\) it is growth (a population that doubles: \(100 \cdot 2^{x}\)); if \(0 < b < 1\) it is decay. At \(x = 3\), \(100 \cdot 2^3 = 100 \cdot 8 = 800\).

⚠️ Common trap: exponential growth (multiplying each step) far outpaces linear growth (adding each step). Don't treat \(2^x\) like \(2x\).

💡 Tip: in \(y = a \cdot b^x\), \(a\) is the starting amount and \(b\) is the growth factor (\(1 + r\) for a rate \(r\)).