Exponents, Logarithms & Radicals

What you need to know

• Exponent rules. $a^ma^n=a^{m+n}$, $(a^m)^n=a^{mn}$ (with fraction exponents this needs $a>0$: $\left((-3)^2\right)^{1/2}=3$, not $-3$), $a^{-n}=\dfrac1{a^n}$, $a^{1/n}=\sqrt[n]a$, and $a^0=1$ (for $a\ne0$). Example: $2^3\cdot2^4=2^7=128$ and $9^{-1/2}=\frac13$.
• Use a common base. Powers with different bases combine only after you rewrite them with the same base. Example: $4^x=2^{2x}$, and $9^x=27$ becomes $3^{2x}=3^3$, so $x=\frac32$.
• Square roots. $\sqrt a\sqrt b=\sqrt{ab}$ when $a$ and $b$ are zero or positive. Also $\sqrt{a^2}=|a|$, not $a$. Example: $\sqrt{(-5)^2}=\sqrt{25}=5$.
• Rationalizing. To clear the roots from the bottom of $\dfrac1{\sqrt a\pm\sqrt b}$, multiply top and bottom by the conjugate $\sqrt a\mp\sqrt b$ (same terms, opposite sign). Example: $\dfrac1{\sqrt3+\sqrt2}=\dfrac{\sqrt3-\sqrt2}{3-2}=\sqrt3-\sqrt2$.
• Denesting. $\sqrt{a+2\sqrt b}=\sqrt m+\sqrt n$ and $\sqrt{a-2\sqrt b}=\sqrt m-\sqrt n$, where $m+n=a$, $mn=b$, and $m\ge n$. Check by squaring. Example: $(\sqrt3\pm\sqrt2)^2=5\pm2\sqrt6$, so $\sqrt{5\pm2\sqrt6}=\sqrt3\pm\sqrt2$.
• Logarithms (rare). $\log_b x=y$ means the same as $b^y=x$. For positive $x$ and $y$: $\log(xy)=\log x+\log y$, $\log x^k=k\log x$, and $\log_b x=\dfrac{\log x}{\log b}$ (top and bottom in the same base). Example: $\log_2 8=3$ because $2^3=8$, and $\log_4 8=\dfrac{\log_2 8}{\log_2 4}=\dfrac32$.
• Comparing huge powers. Take the same root of both. Example: to compare $2^{300}$ and $3^{200}$, write them as $8^{100}$ and $9^{100}$. So $3^{200}$ is bigger.
• Whole-number roots. For a positive integer $N$, $\sqrt[n]{N}$ is an integer exactly when $N$ is a perfect $n$th power. Factor $N$ to check: every exponent must be a multiple of $n$. Example: $216=2^3\cdot3^3$, so $\sqrt[3]{216}=6$; but $72=2^3\cdot3^2$, so $\sqrt[3]{72}$ is not an integer.

How AMC 10 tests it

• Problems 1–8: simplify $\dfrac{2^{2020}+2^{2022}}{2^{2021}}$ or find $8^{2/3}$ or $(\sqrt2)^6$. The trick is factoring out the smallest power.
• Problems 5–12: solve $2^{x+1}=8^{x-2}$ (match the bases) or $\sqrt{x+5}-\sqrt x=1$ (get a root alone, square, check).
• Problems 8–15: "which is largest" among $2^{40},3^{30},5^{20}$, or the number of digits of $2^{k}5^{m}$.
• Problems 10–18: nested roots like $\sqrt{6+\sqrt{6+\cdots}}$, or sums of $\frac1{\sqrt k+\sqrt{k+1}}$ that you rationalize and then telescope (almost everything cancels).
• Logarithms show up only now and then, usually late in the paper, as an exponent equation in disguise.

Standard approaches

1. Write everything with one base and set the exponents equal. Example: $8^{x-2}=2^{3x-6}$, so $2^{x+1}=8^{x-2}$ means $x+1=3x-6$, and $x=\frac72$.
2. In a sum of powers, factor out the smallest power: $2^n+2^{n+2}=2^n(1+4)$. Example: $\dfrac{2^{2020}+2^{2022}}{2^{2021}}=\dfrac{2^{2020}\cdot5}{2^{2021}}=\dfrac52$.
3. For an equation with square roots, get one root alone, square, repeat if needed, and test every answer in the original equation. Example: $\sqrt{x+5}=1+\sqrt x$ squares to $x+5=1+2\sqrt x+x$, so $\sqrt x=2$ and $x=4$. Check: $3-2=1$.
4. For nested or never-ending roots, either guess a denested form $\sqrt m\pm\sqrt n$, or set $x$ equal to the whole expression and square. Example: if $x=\sqrt{6+\sqrt{6+\cdots}}$, then $x^2=6+x$, so $x=3$ (the other root, $-2$, is negative).
5. To compare sizes, take a common power or root, or compare logarithms. Example: the $10$th roots of $2^{40},3^{30},5^{20}$ are $16,27,25$, so $3^{30}$ is largest.

Worked example

Let $x$ be a real number with $4^x+4^{-x}=34$. What is $8^x+8^{-x}$?

(A) $102$ (B) $162$ (C) $198$ (D) $204$ (E) $216$

Let $t=2^x+2^{-x}$. Both terms are positive, so $t$ is positive. Squaring uses $(2^x)^2=4^x$ and $2^x\cdot2^{-x}=1$:
$$t^2=4^x+2+4^{-x}=34+2=36,\qquad t=6 .$$
Now cube. Let $u=2^x$ and $v=2^{-x}$, so $u+v=6$ and $uv=1$. Since $8^x=(2^x)^3=u^3$ and $8^{-x}=v^3$, we want $u^3+v^3$. The identity $u^3+v^3=(u+v)^3-3uv(u+v)$ gives
$$8^x+8^{-x}=t^3-3t=216-18=198 .$$
The answer is $\boxed{\textbf{(C)}\ 198}$.

Pitfalls

• Writing $\sqrt{a+b}=\sqrt a+\sqrt b$ or $(a+b)^2=a^2+b^2$. Example: $\sqrt{9+16}=5$, not $3+4=7$.
• Adding exponents across different bases. Example: $2^3\cdot3^2=72$, not $6^5$.
• Keeping an extra root that appeared when you squared. Example: squaring $\sqrt x=x-2$ gives $x=(x-2)^2$, so $x=1$ or $x=4$. But $x=1$ gives $1=-1$, so only $x=4$ works.
• Forgetting $\sqrt{x^2}=|x|$, which loses a negative solution or causes a sign error. Example: $\sqrt{x^2}=3$ has two solutions, $x=3$ and $x=-3$.

Traps that recur

• Counting 2014 powers of 2 (including 2^0 or 2^2014) instead of 2^1 through 2^2013, which gives 280. (2014 AMC 10A #25)
• For the fifth root, dividing 312 by 5 and taking the leading digit of the fifth root of 7.77, which ignores that 312 is not a multiple of 5. (2021 AMC Fall 10B #19)
• Guessing that x and y need not be multiples of 7, or splitting 13 unevenly (like 1 and 12) and reporting a larger sum such as 7 * 145. (2024 AMC 10B #13)
• Ignoring the 2-power terms and answering 3^4 = 81 exactly, when the fraction is strictly less than 81 because 2^100 = 16 2^96 is much smaller than 81 2^96. (2018 AMC 10A #14)