Integer Exponents, Scientific Notation and Roots

Integer Exponents, Scientific Notation and Roots

This topic covers integer exponents, scientific notation, and roots. Understanding these concepts is important in mathematics, especially for solving problems in natural and engineering sciences. You will learn how to manipulate exponents, express large and small numbers in scientific notation, and calculate roots.

Integer Exponents

Definition of Integer Exponents

An integer exponent indicates how many times a base number is multiplied by itself. For example, in aⁿ, a is the base and n is the exponent. If n is a positive integer, then:

aⁿ = a × a × a × ... (n times)

For example, 3⁴ = 3 × 3 × 3 × 3 = 81.

Zero and Negative Exponents

The zero exponent rule states that any non-zero number raised to the power of zero equals one:

a⁰ = 1 (for a ≠ 0)

For negative exponents, the rule is:

a^-n = 1/aⁿ

For example, 2^-3 = 1/2³ = 1/8.

Rules for Exponents

There are several important rules for working with exponents:

• Product Rule: a^m × aⁿ = a^m+n
• Quotient Rule: a^m / aⁿ = a^m-n (a ≠ 0)
• Power of a Power Rule: (a^m)ⁿ = a^mn
• Power of a Product Rule: (ab)ⁿ = aⁿbⁿ
• Power of a Quotient Rule: (a/b)ⁿ = aⁿ/bⁿ (b ≠ 0)

Worked Example

Calculate (2³ × 2⁴) / 2².

Using the product and quotient rules:

(23 × 24) / 22 = 23+4 / 22 = 27 / 22 = 27-2 = 25 = 32

Scientific Notation

Definition of Scientific Notation

Scientific notation is a way of expressing very large or very small numbers. It is written as:

c × 10ⁿ

where 1 ≤ c < 10 and n is an integer. For example, the number 300,000 can be written as 3 × 10⁵.

Converting to Scientific Notation

To convert a number to scientific notation:

1. Move the decimal point to the right of the first non-zero digit.
2. Count how many places you moved the decimal point. This number becomes the exponent.
3. If you moved the decimal to the left, the exponent is positive; if to the right, it is negative.

Worked Example

Convert 0.000215 to scientific notation.

Move the decimal point 4 places to the right:

0.000215 = 2.15 × 10-4

Calculating with Scientific Notation

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents:

(3 × 105) × (2 × 10-4) = (3 × 2) × 105 + (-4) = 6 × 101 = 6 × 10

Roots and Surds

Definition of Roots

The square root of a number a is a number b such that b² = a. The principal square root is denoted as √a.

Calculating Roots

To calculate the square root, use a calculator or apply the definition. For example:

√9 = 3

Worked Example

Calculate √(16 × 25).

Using the product rule for square roots:

√(16 × 25) = √16 × √25 = 4 × 5 = 20

Summary

• Integer exponents indicate repeated multiplication.
• Scientific notation expresses large/small numbers compactly.
• Roots are the inverse operation of exponents.

Check your understanding

1. Convert 0.00456 to scientific notation.
2. Calculate 4³ × 4^-1.
3. What is the principal square root of 49?
4. Express 150,000 in scientific notation.