Powers & Roots
What Are Powers (Indices)?
Powers are a shorthand way to show repeated multiplication of the same number. Instead of writing 6 × 6 × 6, we write 6³. They're incredibly useful for making calculations cleaner and understanding how numbers grow exponentially.
Base: The main number being multiplied. In 6³, the base is 6.
Index (Power/Exponent): The small raised number showing how many times to multiply the base by itself. In 6³, the index is 3.
Examples:
6¹ = 6 (the base appears once)
6² = 6 × 6 = 36 (called "six squared")
6³ = 6 × 6 × 6 = 216 (called "six cubed")
Special Rules You Must Know
- Any number to the power of 0 equals 1
Examples: 3⁰ = 1, 100⁰ = 1, even (-5)⁰ = 1. This is true for any non-zero number.
- Any number to the power of 1 equals itself
Examples: 3¹ = 3, 7¹ = 7. The power of 1 is invisible in everyday maths.
What Are Square Roots?
Square roots are the reverse of squaring a number. If you square 5 to get 25, the square root of 25 brings you back to 5. They're essential for solving many geometry and algebra problems.
Square Root: A number that when multiplied by itself equals the original number. The square root of 25 is 5 because 5 × 5 = 25.
Important Facts
- Every positive number has TWO square roots — one positive and one negative.
Why? Because 5² = 25 AND (-5)² = 25. Both work!
- The √ symbol refers only to the positive square root.
So √25 = 5 (not ±5)
- To show both roots, use the ± symbol:
√25 = ±5 means "plus or minus 5"
- Negative numbers do NOT have a real square root.
You can't square any real number and get a negative answer, so √(-9) is impossible in basic maths.
Real-world connection: If a square has an area of 100 cm², the length of each side is √100 = 10 cm. This is why square roots are important in geometry.
What Are Cube Roots?
Cube roots are like square roots, but for cubes. They find the number that when multiplied by itself three times gives you the original number.
Cube Root: A number that when cubed (multiplied by itself three times) equals the original number. The cube root of 125 is 5 because 5 × 5 × 5 = 125.
Key Difference from Square Roots
- Each number has only ONE real cube root, not two. This is because (-5)³ = -125, not 125. The sign matters for cube roots!
- Both positive and negative numbers have cube roots.
Example: ³√(-8) = -2 because (-2)³ = -8
- The notation ³√ means "cube root of"
So ³√125 = 5
Common Cube Roots to Remember:
1³ = 1, so ³√1 = 1
2³ = 8, so ³√8 = 2
3³ = 27, so ³√27 = 3
4³ = 64, so ³√64 = 4
5³ = 125, so ³√125 = 5
What Are nth Roots?
How Do I Estimate a Root?
What Are Reciprocals?