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Rules of Exponents and Scientific Notation

Use the product, quotient and power rules, zero, negative and fractional exponents, and scientific notation.

Practice this skill 25 min Rules of exponents Leer en español
Key idea

What you'll learn

Exponents are a shorthand for repeated multiplication. In this lesson you will learn all the exponent rules the GED uses: the product, quotient and power rules, zero and negative exponents, fractional (rational) exponents, and scientific notation for very large and very small numbers.

How it shows up on the GED:

  • "Which expression is equivalent to \dfrac{x^7}{x^2}?" or "Simplify (2x^3)^4."
  • Evaluating 4^{-2} or 27^{2/3} — often in the no-calculator section.
  • Science-style word problems: distances in space, sizes of cells and viruses, populations, computer storage — written as 9.3 \times 10^7 or 2 \times 10^{-6}.
  • Later in algebra, exponents appear in polynomials, area and volume formulas, and exponential growth.

The rules all come from one idea: count the factors. If you forget a rule on test day, write out the factors and count — you can always rebuild it.

Words to know

  • Base — the number being multiplied (in 5^3, the base is 5).
  • Exponent (power) — how many times the base is used as a factor (in 5^3, the exponent is 3): 5^3 = 5 \times 5 \times 5 = 125.
  • Squared / cubed — raised to the power 2 / 3 (7^2 = 49, 2^3 = 8).
  • Coefficient — the number in front of a variable (in 6x^2, the coefficient is 6).
  • Reciprocal — "one over" a number (\tfrac{1}{8} is the reciprocal of 8).
  • Square root \sqrt{a} — the number that squares to a (\sqrt{49} = 7).
  • Cube root \sqrt[3]{a} — the number that cubes to a (\sqrt[3]{64} = 4).
  • Rational exponent — a fraction exponent such as \tfrac12 or \tfrac23; it means a root (9^{1/2} = 3).
  • Scientific notation — a number written as c \times 10^n, where 1 \le c < 10 and n is an integer (52{,}000 = 5.2 \times 10^4).
  • Power of ten — 10^n; for example 10^3 = 1000 and 10^{-2} = 0.01.

All Math words in the glossary

What an exponent means (and the sign trap)

An exponent counts repeated factors:

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32
Five boxes each containing 2, joined by multiplication signs: 2 × 2 × 2 × 2 × 2. A bracket below says 5 factors of 2, written 2 to the 5th power, which equals 32 — not 2 × 5 = 10. 2 × 2 × 2 × 2 × 2 = 32 5 factors of 2 2 5 the exponent counts the factors (2⁵ = 32, not 2 × 5 = 10)
  • a^1 = a (one factor): 9^1 = 9.
  • It is not base times exponent: 2^5 \ne 10.
  • Order matters: 2^5 = 32 but 5^2 = 25.

Powers of 10 are easy: the exponent counts the zeros. 10^4 = 10{,}000, 10^6 = 1{,}000{,}000.

Negative bases — watch the parentheses.

  • (-2)^4 = (-2)(-2)(-2)(-2) = 16 — the base is -2.
  • -2^4 = -(2 \times 2 \times 2 \times 2) = -16 — the base is just 2; the minus is applied afterwards.
  • An even power of a negative number is positive; an odd power is negative: (-3)^3 = -27.

Order of operations: exponents come before multiplication and addition. 3 \cdot 2^3 = 3 \cdot 8 = 24, not 6^3.

The rules for multiplying, dividing and raising powers

Each rule below works only when the bases are the same (for multiplying and dividing).

RuleIn symbolsExample
Product rulea^m \cdot a^n = a^{m+n}x^4 \cdot x^5 = x^9
Quotient rule\dfrac{a^m}{a^n} = a^{m-n}\dfrac{7^6}{7^2} = 7^4
Power of a power(a^m)^n = a^{mn}(x^3)^4 = x^{12}
Power of a product(ab)^n = a^n b^n(2x)^3 = 8x^3
Power of a quotient\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\left(\dfrac{x}{3}\right)^2 = \dfrac{x^2}{9}

Why the product rule works: x^2 \cdot x^3 = (x \cdot x)(x \cdot x \cdot x) = x^5 — five factors in all, so add.

Product rule picture: x squared is two boxes of x (blue) and x cubed is three boxes of x (red). Side by side they make 5 factors of x, so x squared times x cubed equals x to the 5th power. x × x × x × x × x x 2 x 3 2 + 3 = 5 factors of x x² · x³ = x²⁺³ = x⁵

Coefficients follow normal arithmetic. 3x^2 \cdot 4x^5 = (3 \cdot 4)x^{2+5} = 12x^7. And \dfrac{12x^8}{4x^3} = 3x^5.

Different bases cannot be combined: x^2 \cdot y^3 stays x^2y^3. And adding powers has no shortcut: x^2 + x^3 cannot be simplified, and 2^3 + 2^3 = 16, not 2^6.

Zero, negative and fractional exponents

Zero exponent. For any nonzero a: a^0 = 1. Reason: \dfrac{a^3}{a^3} = a^{3-3} = a^0, and anything divided by itself is 1. Examples: 5^0 = 1, (-8)^0 = 1, (3x)^0 = 1. But 3x^0 = 3 \cdot 1 = 3 (only x is raised to 0).

Negative exponent. a^{-n} = \dfrac{1}{a^n} — it means reciprocal, not negative.

  • 2^{-3} = \dfrac{1}{2^3} = \dfrac18
  • 10^{-2} = \dfrac{1}{100} = 0.01
  • \dfrac{1}{x^{-4}} = x^4 (a negative exponent in the denominator moves up)
  • 5x^{-2} = \dfrac{5}{x^2} (only x moves; the 5 stays)

Pattern: 2^3 = 8, 2^2 = 4, 2^1 = 2, 2^0 = 1, 2^{-1} = \tfrac12, 2^{-2} = \tfrac14 — each step down divides by 2.

Bar chart of powers of 2: 2 cubed = 8, 2 squared = 4, 2 to the 1st = 2, 2 to the 0 = 1, 2 to the -1 = 1/2, 2 to the -2 = 1/4. Each step to the right lowers the exponent by 1 and divides the value by 2. 8 2 3 ÷2 4 2 2 ÷2 2 2 1 ÷2 1 2 0 ÷2 1 2 2 −1 ÷2 1 4 2 −2 Each step right: exponent goes down 1, value is divided by 2

Fractional (rational) exponents. The denominator is a root, the numerator is a power:

a^{\frac1n} = \sqrt[n]{a} \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m
  • 25^{1/2} = \sqrt{25} = 5
  • 8^{1/3} = \sqrt[3]{8} = 2
  • 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4
  • 16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8
  • 9^{-1/2} = \dfrac{1}{\sqrt9} = \dfrac13 (negative and fraction: reciprocal, then root)

Scientific notation

Scientific notation writes a number as

c \times 10^n, \qquad 1 \le c < 10,\ n \text{ an integer}

Big number → scientific notation: move the decimal point left until one nonzero digit is in front; the number of moves is a positive exponent. 52{,}000 = 5.2 \times 10^4 (4 moves).

Small number → scientific notation: move the point right; the exponent is negative. 0.00045 = 4.5 \times 10^{-4} (4 moves).

Moving the decimal point. Top: 52,000. becomes 5.2 after the point hops 4 places left, so 52,000 = 5.2 × 10 to the 4th. Bottom: 0.00045 becomes 4.5 after the point hops 4 places right, so 0.00045 = 4.5 × 10 to the -4. 5 2 0 0 0 4 hops left = 5.2 × 10⁴ big number → positive exponent 0 0 0 0 4 5 4 hops right = 4.5 × 10⁻⁴ small number → negative exponent red dot = old decimal point, blue dot = new decimal point

Back to standard form: positive n → move the point right n places; negative n → move left. 3.07 \times 10^5 = 307{,}000; 6.1 \times 10^{-3} = 0.0061.

Multiplying: multiply the fronts, add the exponents, then fix the form. (3 \times 10^4)(5 \times 10^6) = 15 \times 10^{10} = 1.5 \times 10^{11}.

Dividing: divide the fronts, subtract the exponents. \dfrac{8 \times 10^9}{2 \times 10^3} = 4 \times 10^6.

Comparing: first compare exponents (bigger exponent = bigger number, for positive values); if equal, compare the fronts. 2.1 \times 10^{-3} > 9.8 \times 10^{-4} because -3 > -4.

Formula

Exponent rules (memorize — not on the formula sheet)

The GED formula sheet does not list the exponent rules. Memorize them (for a \ne 0, b \ne 0):

a^m \cdot a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn} (ab)^n = a^n b^n \qquad a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m
  • a, b — bases
  • m, n — exponents (can be positive, negative, zero or fractions)

Scientific notation: c \times 10^n with 1 \le c < 10.

Worked example

Example 1 (easy): product and quotient rules

Simplify \dfrac{x^4 \cdot x^5}{x^3}.

  1. Numerator, product rule: x^4 \cdot x^5 = x^{4+5} = x^9.
  2. Quotient rule: \dfrac{x^9}{x^3} = x^{9-3} = x^6.
Fraction with nine factors of x on top (x to the 4th times x to the 5th) and three factors of x on the bottom (x cubed). Three x's on top are crossed out with the three x's on the bottom, leaving six factors of x: the answer is x to the 6th. x x x x x x x x x x x x x 4 x 5 = x 6 9 factors on top, 3 cancel with the 3 on the bottom: 9 − 3 = 6 green slashes: x/x = 1

Answer: x^6.

Check: with x = 2: \dfrac{16 \cdot 32}{8} = \dfrac{512}{8} = 64 = 2^6. ✓

Worked example

Example 2: zero, negative and fractional exponents

Evaluate (a) 2^{-3}, (b) 5^0, (c) 16^{3/4}.

  1. (a) 2^{-3} = \dfrac{1}{2^3} = \dfrac18 = 0.125.
  2. (b) Any nonzero number to the 0 power is 1: 5^0 = 1.
  3. (c) Fourth root first: \sqrt[4]{16} = 2 (since 2^4 = 16). Then cube: 2^3 = 8.
Two flow diagrams. Top: 2 to the -3 means 1 over 2 cubed, which is 1/8. Bottom: 16 to the 3/4 power — first take the 4th root of 16 to get 2 (the denominator 4 is the root), then cube 2 to get 8 (the numerator 3 is the power). 2 −3 reciprocal 1 2³ 2³ = 8 1 8 16 3/4 4th root (denominator 4) 2 cube (numerator 3) 8 2⁴ = 16, so the 4th root of 16 is 2; 2³ = 8

Answers: \tfrac18, 1, 8.

Check (c): 8^4 = 4096 and 16^3 = 4096, so 8 = 16^{3/4}. ✓

Worked example

Example 3: several rules at once

Simplify (2a^3b^{-2})^3 and write it with positive exponents.

  1. Cube every factor: 2^3 \cdot (a^3)^3 \cdot (b^{-2})^3.
  2. 2^3 = 8; (a^3)^3 = a^9; (b^{-2})^3 = b^{-6}.
  3. Result: 8a^9b^{-6} = \dfrac{8a^9}{b^6}.
Table showing (2a³b⁻²)³ as three copies of the factors 2, a cubed and b to the -2 multiplied together. Column results: 2 × 2 × 2 = 8; a³ · a³ · a³ = a⁹ (3 + 3 + 3 = 9); b⁻² · b⁻² · b⁻² = b⁻⁶ (−2 − 2 − 2 = −6). Final answer 8a⁹/b⁶. 2 2 2 a³ a³ a³ b⁻² b⁻² b⁻² copy 1 copy 2 copy 3 2×2×2 = 8 a³⁺³⁺³ = a⁹ b⁻²⁻²⁻² = b⁻⁶ (2a³b⁻²)³ = 8a⁹b⁻⁶ = 8a⁹/b⁶

Answer: \dfrac{8a^9}{b^6}.

Check: with a = 1, b = 2: (2 \cdot 1 \cdot \tfrac14)^3 = (\tfrac12)^3 = \tfrac18, and \tfrac{8}{64} = \tfrac18. ✓

Worked example

Example 4 (word problems): scientific notation

(a) Earth's mass is about 6.0 \times 10^{24} kg and the Moon's is about 7.3 \times 10^{22} kg. About how many times heavier is Earth?

(b) A 2-terabyte drive holds 2 \times 10^{12} bytes. Each photo uses 4 \times 10^6 bytes. How many photos fit?

  1. (a) Divide: \dfrac{6.0}{7.3} \approx 0.822 and 10^{24-22} = 10^2. So 0.822 \times 100 \approx 82.
  2. (b) Divide: \dfrac{2}{4} = 0.5 and 10^{12-6} = 10^6. So 0.5 \times 10^6 = 5 \times 10^5 = 500{,}000 photos.
Powers-of-ten ruler from 10 to the 20th to 10 to the 25th kilograms. The Moon's mass, 7.3 × 10²² kg, sits just below 10²³. Earth's mass, 6.0 × 10²⁴ kg, sits just below 10²⁵. They are almost 2 powers of ten apart, so Earth is about 82 times heavier (6.0 ÷ 7.3 × 100 ≈ 82). 10 20 10 21 10 22 10 23 10 24 10 25 Moon 7.3 × 10²² Earth 6.0 × 10²⁴ × about 82 mass (kg), each big tick = 10 times the one before

Answers: (a) about 82 times; (b) 500,000 photos.

Check (b): 500{,}000 \times 4{,}000{,}000 = 2{,}000{,}000{,}000{,}000 = 2 \times 10^{12}. ✓

Worked example

Example 5 (GED-hard, work backwards): find the missing exponent

(a) Find n if 3^n \cdot 3^4 = 3^{10}. (b) Find x if 2^x = \dfrac{1}{32}.

  1. (a) Product rule: 3^{n+4} = 3^{10}. Same base, so the exponents are equal: n + 4 = 10, so n = 6.
  2. (b) Write 32 as a power of 2: 32 = 2^5. Then \dfrac{1}{32} = \dfrac{1}{2^5} = 2^{-5}.
  3. So 2^x = 2^{-5}, which means x = -5.
A strip of 10 boxes, each containing 3, showing 3 to the 10th. The first 6 boxes are bracketed as 3 to the n with n = 6 and the last 4 boxes as 3 to the 4th, so n + 4 = 10 gives n = 6. 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 3 n 3 4 all together: 310 n + 4 = 10, so n = 6

Answers: n = 6; x = -5.

Check: 3^6 \cdot 3^4 = 729 \cdot 81 = 59{,}049 = 3^{10} ✓, and 2^{-5} = \tfrac{1}{32} ✓.

Calculator tip

TI-30XS MultiView tips

  • Any power: use the ^ key. 3^4: 3 ^ 4 enter → 81. Press the right arrow to leave the exponent before typing more.
  • Squares: the x² key. Square roots: 2nd then x² (√).
  • Other roots: type the root index first, then 2nd ^ (ˣ√), then the number: 3 2nd ^ 27 enter → 3.
  • Negative bases: use parentheses and the (−) key: ( (−) 2 ) ^ 4 → 16. Without parentheses the calculator computes -2^4 = -16.
  • Negative or fractional exponents: 2 ^ (−) 3 enter → 0.125; press toggle to see \tfrac18. For 27^{2/3}, type the exponent with the n/d key or as (2÷3).
  • Scientific notation: the ×10ⁿ key enters a power of ten: 1.5 ×10ⁿ 8 gives 1.5 \times 10^8. If you type × 10 ^ instead, press the right arrow to leave each exponent and wrap each number in parentheses when dividing: ( 1.5 × 10 ^ 8 ▶ ) ÷ ( 3 × 10 ^ 5 ▶ ) enter.

When not to use it: "Which expression is equivalent to ...?" questions with variables cannot be done on the calculator — you need the rules. Small powers like 2^5 or \sqrt[3]{27} are expected without a calculator.

Common mistake

Common mistakes

  • Multiplying base and exponent. Wrong: 3^4 = 12. Right: 3 \times 3 \times 3 \times 3 = 81.
  • Mixing up the product and power rules. Wrong: x^5 \cdot x^3 = x^{15}. Right: add when multiplying powers (x^8); multiply only for (x^5)^3 = x^{15}.
  • Forgetting the coefficient. Wrong: (2x^3)^4 = 2x^{12}. Right: 2^4x^{12} = 16x^{12}.
  • Thinking a negative exponent makes a negative number. Wrong: 4^{-2} = -16. Right: 4^{-2} = \tfrac{1}{16}.
  • Sign trap with negative bases. Wrong: -3^2 = 9. Right: -3^2 = -9, while (-3)^2 = 9.
  • Scientific notation errors. Wrong: 93{,}000{,}000 = 93 \times 10^6 (front not between 1 and 10) or 9.3 \times 10^6 (counted zeros). Right: 9.3 \times 10^7.
Test strategy

GED test strategy

  • Test with a number. For "which expression is equivalent?", pick x = 2 (avoid 0 and 1), evaluate the original and each choice, and keep the one that matches.
  • Eliminate by sign and size. A negative exponent on a positive base gives a positive number less than 1. Choices that are negative or large are out.
  • Rewrite with a common base. 4 = 2^2, 8 = 2^3, 27 = 3^3, 1000 = 10^3. Then just work with exponents.
  • Scientific notation answers: check that the front number is between 1 and 10, and that big numbers have positive exponents and small numbers negative ones.
  • Read the units. Word problems may give seconds but ask for minutes, or meters but ask for kilometers.
  • Time: simple evaluations should take under a minute. Leave long simplifications for the end of a section if you are short on time.

Check yourself

Rules of exponentsNo calculator

What is the value of 3^4?

Answer choices
Hint

The exponent tells you how many times to use the base as a factor.

Check yourself

Rules of exponentsNo calculator

Which expression is equivalent to x^5 \cdot x^3?

Answer choices
Hint

Write out each power as repeated factors and count how many x's you have in total.

Summary

  • An exponent counts repeated factors: a^n = a \times a \times \cdots \times a (n times).
  • Same base: multiply → add exponents; divide → subtract; power of a power → multiply.
  • a^0 = 1, a^{-n} = \dfrac{1}{a^n} (reciprocal, not negative), and a^{m/n} = (\sqrt[n]{a})^m.
  • Parentheses matter: (-2)^4 = 16 but -2^4 = -16.
  • Scientific notation is c \times 10^n with 1 \le c < 10; multiply/divide the fronts and add/subtract the exponents.
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