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Math foundations
SAT math exponents made simple: powers, rules, and practice
Understand what an exponent counts, learn the rules for combining powers, and avoid common traps with zero, negative signs, and negative exponents.
SPSAT Practice Editorial8 min read
The short version
What you'll take away
- An exponent tells how many times to use the base as a factor: 3⁴ means 3 × 3 × 3 × 3.
- When multiplying powers with the same base, add exponents; when dividing, subtract them.
- Zero and negative exponents have precise meanings, and parentheses determine whether a negative sign is part of the base.
In this article
An exponent counts repeated factors
In an expression like \(\displaystyle 3^4\), the 3 is the base and the small raised 4 is the exponent. It means multiply four 3s: \(\displaystyle 3^4 = 3 \times 3 \times 3 \times 3 = 81\). The exponent counts factors; it does not tell you to multiply 3 by 4.
Read \(\displaystyle 5^2\) as “five squared”: \(\displaystyle 5 \times 5 = 25\). Read \(\displaystyle 2^3\) as “two cubed”: \(\displaystyle 2 \times 2 \times 2 = 8\). Those names are shortcuts; the repeated multiplication is the meaning.
The study lab
The exponent tells how many factors to use
Each repeated tile is one copy of the base; multiply the factors to get the power.
23==8
32==9
53==125
Use the rules when the bases match
When multiplying powers with the same base, keep the base and add the exponents. This works because you are joining repeated factors: \(\displaystyle 2^3 \times 2^4 = (2 \times 2 \times 2)(2 \times 2 \times 2 \times 2) = 2^7 = 128\).
When dividing powers with the same nonzero base, keep the base and subtract the exponents. Matching factors cancel: \(\displaystyle 5^6 / 5^2 = 5^{6-2} = 5^4 = 625\).
- Multiply same bases: \(\displaystyle a^m \times a^n = a^{m+n}\).
- Divide same nonzero bases: \(\displaystyle a^m / a^n = a^{m-n}\).
- Do not add exponents when the bases are different: \(\displaystyle 2^3 \times 3^3\) is not \(\displaystyle 6^6\).
A power raised to a power multiplies exponents
In \(\displaystyle (3^2)^4\), the inside power has four copies of \(\displaystyle 3^2\): \(\displaystyle 3^2 \times 3^2 \times 3^2 \times 3^2\). That is eight 3s altogether, so multiply the exponents: \(\displaystyle (3^2)^4 = 3^{2 \times 4} = 3^8\).
A power of a product applies to each factor: \(\displaystyle (2 \times 5)^2 = 2^2 \times 5^2 = 100\). Keep the parentheses so you can see exactly what is being squared.
Zero and negative exponents
Any nonzero number raised to the zero power equals 1: \(\displaystyle 7^0 = 1\), and \(\displaystyle (-4)^0 = 1\). One way to see why is to use the quotient rule: \(\displaystyle 7^3 / 7^3 = 7^{3-3} = 7^0\), while any nonzero number divided by itself is 1.
A negative exponent means take the reciprocal: \(\displaystyle 2^{-3} = \frac{1}{2^3} = \frac18\). It does not make the answer negative. The base must not be zero, because division by zero is undefined.
Parentheses decide what a negative sign belongs to
Compare \(\displaystyle (-3)^2\) and \(\displaystyle -3^2\). In the first expression, the base is negative 3, so \(\displaystyle (-3)(-3) = 9\). In the second, the exponent applies to 3 first, and the negative sign stays outside: \(\displaystyle -(3^2) = -9\).
For an odd exponent, a negative base stays negative: \(\displaystyle (-2)^3 = -8\). For an even exponent, its product is positive: \(\displaystyle (-2)^4 = 16\). Use the parentheses to identify the full base before calculating.
- \(\displaystyle (-3)^2 = 9\): the base includes the negative sign.
- \(\displaystyle -3^2 = -9\): exponent first, then the leading negative sign.
- A negative base to an odd power is negative; to an even power it is positive.
Practice: try these before checking the answers
Write out a few repeated factors if a rule feels abstract. For sign questions, mark the base with parentheses before you start.
- 1. Evaluate \(\displaystyle 4^3\).
- 2. Simplify \(\displaystyle 2^3 \times 2^5\).
- 3. Simplify \(\displaystyle 5^6 / 5^4\).
- 4. Simplify \(\displaystyle (2^3)^2\).
- 5. Evaluate \(\displaystyle 10^0\).
- 6. Rewrite \(\displaystyle 2^{-3}\) using a positive exponent.
- 7. Evaluate both \(\displaystyle (-3)^2\) and \(\displaystyle -3^2\).
A quick exponent check for SAT Math
Before calculating, identify the base and exponent and check whether parentheses include a negative sign. Then choose a rule only if its conditions fit: same base for multiplying or dividing powers, and a power raised to a power for multiplying exponents.
Finally, estimate the size. A positive base greater than 1 grows as its exponent increases; a positive base between 0 and 1 gets smaller. Rewriting a negative exponent as a reciprocal can make that size easier to see.
- Expand a small power to confirm what the exponent counts.
- Keep parentheses around a negative base when it is raised to a power.
- Use a quick estimate to catch a misplaced exponent or sign.
Sources & further reading
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