Exponents and scientific notation

Start with repeated factors

For a positive integer exponent, a^n means n factors equal to a. Thus 3^4 means 3 × 3 × 3 × 3, not 3 × 4. The caret symbol in these lessons means 'raised to the power of.' When multiplying powers of the same base, combine the factors: a^m × a^n = a^(m+n). This rule concerns multiplication, not addition. In general, a^m + a^n cannot be simplified by adding the exponents.

Worked example: simplify without a calculator

Consider (2^5 × 2^3)/2^4. The numerator contains eight factors of two, so it is 2^8. Dividing by four factors of two leaves 2^(8-4) = 2^4 = 16. A separate expression, (2^3)^2, repeats the group 2^3 twice and equals 2^6 = 64. The exponent on a power is multiplied by the outer exponent. These two operations are easy to confuse: multiplying equal-base powers adds exponents; raising a power to another power multiplies them.

Zero, negatives and parentheses

For nonzero a, a^0 = 1 and a^(-n) = 1/a^n. A negative exponent indicates a reciprocal, not a negative answer. For example, 5^(-2) = 1/25. Parentheses determine what is being raised: (-4)^2 = 16 because the negative number is the base, while -4^2 = -16 under standard order of operations because the square is evaluated before the leading negative sign. The expression 0^(-1) is undefined, and the rules here do not assign a value to 0^0.

Scientific notation records scale

A positive number in normalized scientific notation has the form a × 10^n, with 1 ≤ a < 10 and integer n. The number 0.000072 becomes 7.2 × 10^(-5); 4,600,000 becomes 4.6 × 10^6. For their product, multiply coefficients and add exponents: 7.2 × 4.6 × 10^1 = 33.12 × 10^1 = 3.312 × 10^2, or 331.2. The intermediate coefficient 33.12 is valid algebraically but is not normalized scientific notation until it is adjusted. Negative numbers use the same size convention with a negative coefficient: -0.0038 = -3.8 × 10^(-3). The minus sign indicates sign, while the exponent describes scale; these are separate pieces of information.

Adding values requires a shared scale

To add 3.4 × 10^5 and 8 × 10^4, rewrite the second as 0.8 × 10^5. The sum is 4.2 × 10^5. Do not multiply the coefficients or add the powers for this addition. When a problem contains measurements, significant-figure instructions may affect the final reported precision; the arithmetic examples here treat the stated numbers as exact unless told otherwise.

• Check whether the operation is addition, multiplication, division or a power of a power.
• Attach a negative exponent to the correct base, using parentheses when needed.
• Estimate the order of magnitude before accepting a long decimal answer.

Sources & further reading