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Exponents, Logarithms & Radicals

Exponent rules, radicals, nested radicals, rationalizing, logarithms (rare)

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Exponents, Logarithms & RadicalsAMC 10 Step by Step

Name: Date: October 28, 2026

all 36 problems, easiest first

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

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 $x=\frac72$ .
In a sum of powers, factor out the smallest power: $2^n+2^{n+2}=2^n(1+4)$ . Example: $\dfrac{2^{2020}+2^{2022}}{2^{2021}}=\dfrac{2^{2020}\cdot5}{2^{2021}}=\dfrac52$ .
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: $3-2=1$ .
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).
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)

1.2023 AMC 10A #5
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$ ?
1. A) $14$
2. B) $15$
3. C) $16$
4. D) $17$
5. E) $18$
2.2021 AMC 10A #1
What is the value of $(2^2-2)-(3^2-3)+(4^2-4)?$
1. A) $1$
2. B) $2$
3. C) $5$
4. D) $8$
5. E) $12$
3.2021 AMC Fall 10B #5
Let $n=8^{2022}$ . Which of the following is equal to $\frac{n}{4}?$
1. A) $4^{1010}$
2. B) $2^{2022}$
3. C) $8^{2018}$
4. D) $4^{3031}$
5. E) $4^{3032}$
4.2021 AMC 10B #2
What is the value of $\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}$ ?
1. A) $0$
2. B) $4\sqrt{3}-6$
3. C) $6$
4. D) $4\sqrt{3}$
5. E) $4\sqrt{3} + 6$
5.2019 AMC 10A #1
What is the value of $2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?$
1. A) $0$
2. B) $1$
3. C) $2$
4. D) $3$
5. E) $4$
6.2018 AMC 10A #1
What is the value of $\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?$
1. A) $\frac58$
2. B) $\frac{11}7$
3. C) $\frac85$
4. D) $\frac{18}{11}$
5. E) $\frac{15}8$
7.2016 AMC 10B #1
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \tfrac{1}{2}$ ?
1. A) $1$
2. B) $2$
3. C) $\frac{5}{2}$
4. D) $10$
5. E) $20$
8.2016 AMC 10A #2
For what value $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$ ?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
9.2015 AMC 10B #1
What is the value of $2-(-2)^{-2}$ ?
1. A) $-2$
2. B) $\dfrac{1}{16}$
3. C) $\dfrac{7}{4}$
4. D) $\dfrac{9}{4}$
5. E) $6$
10.2015 AMC 10A #1
What is the value of $(2^0-1+5^2+0)^{-1}\times5?$
1. A) $-125$
2. B) $-120$
3. C) $\frac{1}{5}$
4. D) $\frac{5}{24}$
5. E) $25$
11.2014 AMC 10B #2
What is $\frac{2^3 + 2^3}{2^{-3} + 2^{-3}}$ ?
1. A) $16$
2. B) $24$
3. C) $32$
4. D) $48$
5. E) $64$
12.2013 AMC 10A #8
What is the value of $\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?$
1. A) $-1$
2. B) $1$
3. C) $\frac{5}{3}$
4. D) $2013$
5. E) $2^{4024}$
13.2008 AMC 10B #3
Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}$ ?
1. A) $x^{1/6}$
2. B) $x^{1/4}$
3. C) $x^{3/8}$
4. D) $x^{1/2}$
5. E) $x$
14.2004 AMC 10B #5
In the expression $c\cdot a^b-d$ , the values of $a$ , $b$ , $c$ , and $d$ are $0$ , $1$ , $2$ , and $3$ , although not necessarily in that order. What is the maximum possible value of the result?
1. A) $5$
2. B) $6$
3. C) $8$
4. D) $9$
5. E) $10$
15.2002 AMC 10B #1
The ratio $\frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is:
1. A) $1/6$
2. B) $1/3$
3. C) $1/2$
4. D) $2/3$
5. E) $3/2$
16.2002 AMC 10A #1
The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?
1. A) $0.1$
2. B) $0.2$
3. C) $1$
4. D) $5$
5. E) $10$
17.2000 AMC 10 #2
$2000(2000^{2000}) = x$ Find x.
1. A) $2000^{2001}$
2. B) $4000^{2000}$
3. C) $2000^{4000}$
4. D) $4,000,000^{2000}$
5. E) $2000^{4,000,000}$
18.2022 AMC 10A #11
Ted mistakenly wrote $2^m\cdot\sqrt{\frac{1}{4096}}$ as $2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ for which these two expressions have the same value?
1. A) $5$
2. B) $6$
3. C) $7$
4. D) $8$
5. E) $9$
19.2018 AMC 10A #7
For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?
1. A) $3$
2. B) $4$
3. C) $6$
4. D) $8$
5. E) $9$
20.2009 AMC 10A #13
Suppose that $P = 2^m$ and $Q = 3^n$ . Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$ ?
1. A) $P^2Q$
2. B) $P^nQ^m$
3. C) $P^nQ^{2m}$
4. D) $P^{2m}Q^n$
5. E) $P^{2n}Q^m$
21.2008 AMC 10A #7
The fraction
$\frac{\left(3^{2008}\right)^2-\left(3^{2006}\right)^2}{\left(3^{2007}\right)^2-\left(3^{2005}\right)^2}$ simplifies to which of the following?
1. A) $1$
2. B) $\frac{9}{4}$
3. C) $3$
4. D) $\frac{9}{2}$
5. E) $9$
22.2007 AMC 10A #9
Real numbers $a$ and $b$ satisfy the equations $3^{a} = 81^{b + 2}$ and $125^{b} = 5^{a - 3}$ . What is $ab$ ?
1. A) $-60$
2. B) $-17$
3. C) $9$
4. D) $12$
5. E) $60$
23.2006 AMC 10B #7
Which of the following is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$ ?
1. A) $-x$
2. B) $x$
3. C) $1$
4. D) $\sqrt{\frac{x}{2}}$
5. E) $x\sqrt{-1}$
24.2006 AMC 10A #6
What non-zero real value for $x$ satisfies $(7x)^{14}=(14x)^7$ ?
1. A) $\frac17$
2. B) $\frac27$
3. C) $1$
4. D) $7$
5. E) $14$
25.2005 AMC 10A #13
How many positive integers $n$ satisfy the following condition:
$\left(130n\right)^{50} > n^{100} > 2^{200} \ \text{?}$
1. A) $0$
2. B) $7$
3. C) $12$
4. D) $65$
5. E) $125$
26.2003 AMC 10B #9
Find the value of $x$ that satisfies the equation $25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.$
1. A) $2$
2. B) $3$
3. C) $5$
4. D) $6$
5. E) $9$
27.2003 AMC 10A #9
Simplify $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}$ .
1. A) $\sqrt{x}$
2. B) $\sqrt[3]{x^{2}}$
3. C) $\sqrt[27]{x^{2}}$
4. D) $\sqrt[54]{x}$
5. E) $\sqrt[81]{x^{80}}$
28.2003 AMC 10B #14
Given that $3^8\cdot5^2=a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$ .
1. A) $25$
2. B) $34$
3. C) $351$
4. D) $407$
5. E) $900$
29.2002 AMC 10A #3
According to the standard convention for exponentiation, $2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.$
If the order in which the exponentiations are performed is changed, how many other values are possible?
1. A) $0$
2. B) $1$
3. C) $2$
4. D) $3$
5. E) $4$
30.2002 AMC 10B #14
The number $25^{64}\cdot 64^{25}$ is the square of a positive integer $N$ . In decimal representation, the sum of the digits of $N$ is
1. A) $7$
2. B) $14$
3. C) $21$
4. D) $28$
5. E) $35$
31.2024 AMC 10B #13
Positive integers $x$ and $y$ satisfy the equation $\sqrt{x} + \sqrt{y} = \sqrt{1183}$ . What is the minimum possible value of $x+y$ ?
1. A) $585$
2. B) $595$
3. C) $623$
4. D) $700$
5. E) $791$
32.2018 AMC 10A #14
What is the greatest integer less than or equal to $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$
1. A) $80$
2. B) $81$
3. C) $96$
4. D) $97$
5. E) $625$
33.2011 AMC 10A #16
Which of the following is equal to $\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$ ?
1. A) $3\sqrt2$
2. B) $2\sqrt6$
3. C) $\frac{7\sqrt2}{2}$
4. D) $3\sqrt3$
5. E) $6$
34.2005 AMC 10B #17
Suppose that $4^a = 5$ , $5^b = 6$ , $6^c = 7$ , and $7^d = 8$ . What is $a \cdot b\cdot c \cdot d$ ?
1. A) $1$
2. B) $\frac{3}{2}$
3. C) $2$
4. D) $\frac{5}{2}$
5. E) $3$
35.2021 AMC Fall 10B #19
Let $N$ be the positive integer $7777\ldots777$ , a $313$ -digit number where each digit is a $7$ . Let $f(r)$ be the leading digit of the $r{ }$ th root of $N$ . What is $f(2) + f(3) + f(4) + f(5)+ f(6)?$
1. A) $8$
2. B) $9$
3. C) $11$
4. D) $22$
5. E) $29$
36.2014 AMC 10A #25
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$ . How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}?$
1. A) $278$
2. B) $279$
3. C) $280$
4. D) $281$
5. E) $282$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.