Exponents and radicals

This chapter applies to all AMC 8/10/12 test takers.

Exponentiation is an operation with two parts: a base and an exponent. When the exponent is a positive integer, it tells you how many times to multiply the base by itself.

$$a^n=\underset{\text{n times}}{\underbrace{a\times a\times \cdots \times a}}$$

Here are the exponent rules you’ll use often on AMC exams.

Rule 1: With multiplication, add the exponents

$$(x^y)(x^z)=x^{(y+z)}$$

$$3^4\times 3^5=3^9$$

Here, both bases are $3$, so we can add the exponents. If you expand each power, you can see why:

$$3^4\times 3^5=(3\times 3\times 3\times 3)\times (3\times 3\times 3\times 3\times 3)=3^9$$

All together, that’s $3$ multiplied by itself $9$ times, and $4+5=9$.

Note that you can only add exponents when the base numbers are the same.

Example: The question below is from 2021 AMC 12A

What is the value of$2^{1+2+3}-(2^1+2^2+2^3)$
A. $0$
B. $50$
C. $52$
D. $54$
E. $57$

Rule 2: With division, subtract the exponents

$$\frac{(x^y)}{(x^z)}=x^{(y-z)}$$

$$\frac{3^6}{3^3}=3^3$$

Again, the bases are the same ($3$), so we subtract the exponents. You can think of this as canceling three factors of $3$ from the numerator and denominator:

$$\frac{3^6}{3^3}=\frac{(3\times 3\times 3\times 3\times 3\times 3)}{(3\times 3\times 3)}=3\times 3\times 3=3^3$$

The exponent becomes $6-3=3$.

Rule 3: A negative exponent means an exponent on the other side of a fraction

$$(x^{-z})=\frac{1}{x^z}$$

$$3^{-4}=\frac{1}{(3^4)}$$

A negative exponent doesn’t mean the number is negative. It tells you to take the reciprocal (flip to the denominator) and make the exponent positive.

You can connect this to Rule 2. For example, since $4-8=-4$, we have $\frac{3^4}{3^8}=3^{-4}$. Expanding shows that four factors of $3$ cancel, leaving:

$$\frac{1}{(3^4)}$$

Rule 4: Any number to the power of 0 is 1

$$x^0=1$$

$$3^0=1$$

This also follows from Rule 2. If you divide equal powers, the exponents subtract to $0$:

$$\frac{3^3}{3^3}=3^{3-3}=3^0=1$$

As a fraction, the numerator equals the denominator, so the value is $1$.

Rule 5: When a number with an exponent is raised to another exponent, multiply the exponents

$$(x^y{)}^z=x^{(yz)}$$

$$(3^3{)}^2=3^6$$

Here, $3^3$ means “three 3’s multiplied,” and squaring means you have two copies of that product:

$(3\times 3\times 3)\times (3\times 3\times 3)$

That’s six factors of $3$, so the result is $3^6$. Multiplying the exponents gives the same exponent: $3\times 2=6$.

Rule 6: When bases are different with the same exponents, multiply the bases and keep the exponent

$$(x^z)(y^z)=(xy{)}^z$$

$$3^3\times 4^3=12^3$$

Expanding shows what’s happening:

$(3\times 3\times 3)\times (4\times 4\times 4)$

Pair each $3$ with a $4$ to make three factors of $12$:

$12\times 12\times 12=12^3$.

Rule 7: An exponent that is a fraction represents a radical

$$\sqrt[z]{x^y}=x^{\frac{y}{z}}$$

It’s often helpful to rewrite radicals as exponents so you can use the exponent rules above. A radical can be written using the root as the denominator and the inside exponent as the numerator:

$$\sqrt{25}=\sqrt[2]{25^1}=25^{\frac{1}{2}}$$

Imaginary numbers

$$(-1{)}^{\frac{1}{2}}=\sqrt{-}1=i$$

The square root of $-1$ is called an imaginary number. No real number multiplied by itself gives a negative result, so we introduce $i$ to represent this value. This idea is developed further in the Complex Numbers chapter of the Advanced Topics part of Unit 1.

Logarithms as exponents

$${\text{log}}_{x}y=z$$

$$x^z=y$$

This will be discussed further in the Logarithmic Functions chapter of the Advanced Topics part of Unit 1. For now, the key idea is that logarithms rewrite an exponential relationship by solving for the exponent.

${\text{Log}}_{x}y=z$ means: “$z$ is the exponent you put on base $x$ to get $y$.” In other words, it’s equivalent to $x^z=y$.

Common themes

• Manipulating exponents will be key to making certain problems easier to work with. Always try to use the rules shown above to simplify exponents.
• You may need to find the unit digit of a number with a very large exponent. To solve, just concern yourself with the unit digit of the base. Other questions may ask for the remainder when divided by 10 or 5. For either of these, only the unit digit is needed.
• Be aware when an exponential function grows rapidly or decays. This can help you delete obviously incorrect answers.
• Exponents are utilized in factorization and can be a key aspect to prime factorization, divisibility rules, and modular arithmetic.