Powers, Roots and Scientific Notation

Introduction

A power (or exponent) tells you how many times to multiply a number by itself. Powers appear in area, compound growth, computer memory and science. Roots undo powers. Scientific notation lets us write huge and tiny numbers compactly.

Powers

2^3 means 2 x 2 x 2 = 8. The number 2 is the base, and 3 is the exponent.

• 5^2 = 25 (5 squared)
• 2^10 = 1,024
• 10^3 = 1,000

Special cases: any non-zero number to the power 0 is 1, so 7^0 = 1. A negative exponent means a reciprocal: 2^-1 = 1/2, and 10^-2 = 1/100 = 0.01.

Roots

A square root reverses squaring. The square root of 144 is 12, because 12 x 12 = 144. The symbol is √. The cube root reverses cubing: the cube root of 27 is 3.

Square roots of non-perfect squares go on forever without repeating, such as √2 ≈ 1.414. They are called irrational numbers.

Laws of exponents

For the same base:

• Multiply: add the exponents. 2^3 x 2^4 = 2^7 = 128.
• Divide: subtract. 2^5 / 2^2 = 2^3 = 8.
• Power of a power: multiply. (2^3)^2 = 2^6 = 64.

Beware: these laws apply to the same base. 2^3 + 2^4 is not 2^7. It is 8 + 16 = 24.

Also, order of operations: in 3 x 2^2, the power comes first, giving 3 x 4 = 12, not 6^2 = 36.

Scientific notation

Write a number as a x 10^n, where a is between 1 and 10.

• 320,000 = 3.2 x 10^5
• 0.00045 = 4.5 x 10^-4

To convert: count how many places the decimal point moves. Moving left makes a positive exponent, right a negative one.

It makes comparing and multiplying easy: (2 x 10^3) x (3 x 10^4) = 6 x 10^7. Calculators and spreadsheets often show 3.2E+05 for 3.2 x 10^5.

Examples: the speed of light is about 3 x 10^8 metres per second. A human hair is about 7 x 10^-5 metres wide.

Orders of magnitude

Each power of ten is one order of magnitude. Thinking in orders of magnitude helps check estimates: a million is 10^6, a billion 10^9. A billion seconds is about 31.7 years, which surprises many people. (1,000,000,000 divided by 31,536,000 seconds per year is about 31.7.)

Try it yourself

Calculate 3^4, the square root of 81 and 2^3 x 2^5. Write 45,000,000 and 0.0062 in scientific notation. Estimate how many seconds there are in a million minutes.

Summary

• A power is repeated multiplication, and a root undoes it.
• Add exponents when multiplying the same base, subtract when dividing and multiply for a power of a power.
• Scientific notation writes numbers as a x 10^n, which makes very large and small numbers easier to handle.