Exponents and Scientific Notation

Grade 8 · Math · Study Guide

Master the rules of exponents and learn to write very large and very small numbers in scientific notation.

An exponent tells you how many times to multiply a base by itself. For example, 2^3 = 2 × 2 × 2 = 8. The base is 2 and the exponent is 3.

There are several important rules for working with exponents. The product rule says that when multiplying powers with the same base, add the exponents: a^m × a^n = a^(m+n). The quotient rule says that when dividing, subtract the exponents: a^m ÷ a^n = a^(m-n). The power rule says that when raising a power to a power, multiply the exponents: (a^m)^n = a^(m×n). Any number raised to the zero power equals 1: a^0 = 1. A negative exponent means take the reciprocal: a^(-n) = 1/a^n.

Scientific notation is a way of writing very large or very small numbers using powers of 10. A number in scientific notation looks like a × 10^n, where 1 ≤ a < 10 and n is an integer. For example, 93,000,000 = 9.3 × 10^7 and 0.0000045 = 4.5 × 10^(-6). To multiply numbers in scientific notation, multiply the decimal parts and add the exponents.

Key terms

Exponent
The number that tells how many times the base is multiplied by itself
Base
The number being multiplied in an exponential expression
Scientific notation
A way to write numbers as a × 10^n, where 1 ≤ a < 10
Negative exponent
Means the reciprocal of the base raised to the positive exponent; a^(-n) = 1/a^n
Zero exponent
Any nonzero number raised to the power of 0 equals 1
Product rule
When multiplying same-base powers, add the exponents: a^m × a^n = a^(m+n)

Flashcards

a^0
Equals 1 for any nonzero value of a
a^(-n)
Equals 1/a^n; a negative exponent means take the reciprocal
Product rule
a^m × a^n = a^(m+n)
Quotient rule
a^m ÷ a^n = a^(m-n)
Power rule
(a^m)^n = a^(m×n)
Scientific notation
a × 10^n where 1 ≤ a < 10; used for very large or very small numbers
Large number in sci. notation
Move decimal left; exponent is positive (e.g., 6,200 = 6.2 × 10^3)
Small number in sci. notation
Move decimal right; exponent is negative (e.g., 0.0062 = 6.2 × 10^-3)

Questions

  1. What is 3^4?

    1. 12
    2. 64
    3. 81
    4. 27
    Show answer

    Answer: 81

    3^4 = 3 × 3 × 3 × 3 = 81.

  2. 2^5 × 2^3 = 2^_____

    Show answer

    Answer: 8

    Using the product rule: 2^5 × 2^3 = 2^(5+3) = 2^8.

  3. The number 450,000 written in scientific notation is 4.5 × 10^5.

    Show answer

    Answer: True

    450,000 = 4.5 × 100,000 = 4.5 × 10^5. The decimal moves 5 places left.

  4. Which is equal to 5^(-2)?

    1. 25
    2. -10
    3. 1/25
    4. -25
    Show answer

    Answer: 1/25

    5^(-2) = 1/5^2 = 1/25. Negative exponents represent reciprocals.

  5. Any nonzero number raised to the zero power equals _____.

    Show answer

    Answer: 1

    By the zero exponent rule, any nonzero base raised to the 0 power equals 1.

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