Powers and roots.
Build a number with a power. Find your way back with a root.
Recall squares to 15 · recall the required cubes · evaluate powers and roots
Powers build. Roots undo.
You already know that multiplication can shorten repeated addition. Powers shorten repeated multiplication: 4 × 4 × 4 becomes 4³. The large number is the base; the small raised number tells us how many equal factors to use.
A root asks us to reverse that multiplication. Instead of asking for the area of a square with a known side, we ask for its side from its area. Read a power forward and its matching root backward before using the symbols in calculations.
A power tells you how many equal factors to multiply. A root reverses that process.
Square ↔ square root
√169 = 13
A square with area 169 cm² has side length 13 cm.
Cube ↔ cube root
∛125 = 5
A cube with volume 125 cm³ has side length 5 cm. The small 3 in ∛ identifies a cube root.
Go backwards with the cube root.
Pause and explainIs 4³ equal to 4 × 3?
No. It means 4 × 4 × 4 = 64. The index counts factors; it is not a multiplier.
Build your recall wall
Recall becomes easier when you learn a pair of facts rather than two unrelated answers. Saying “12 squared is 144” should also remind you that the square root of 144 is 12.
Use the table in small groups. Cover a result, say the multiplication that produces it, then uncover and check. Switch to the roots view and work backward. If you forget a value, reconstruct it from multiplication instead of learning a disconnected digit string.
Recall these powers and their corresponding roots without a calculator. Learning the pairs together makes both directions easier.
Squares: 1 to 15
Cubes: 1, 2, 3, 4, 5 and 10
Cover the bold values and test your recall. Then switch to roots.
Pause and explainHow could knowing 7² = 49 help with √49?
The square root reverses the square, so the non-negative number whose square is 49 is 7.
Calculate with other powers and roots
A calculation may mix powers, roots and ordinary operations. Treat each power or root as a small calculation of its own before combining the results. This keeps several operations from becoming one confusing step.
For example, in 3² + ∛8, first find 3² = 9 and ∛8 = 2, then add to get 11. Negative signs need careful brackets: the square of a negative number is positive, while its cube remains negative.
In 2⁵, 2 is the base and 5 is the index (or exponent). 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The fourth root of 81 is 3 because 3⁴ = 81.
Calculate 5² × ∛8.
- Square first: 5² = 25.
- Find the cube root: ∛8 = 2.
- Multiply: 25 × 2 = 50.
Evaluate ∜81 + √0.49 − ∛(−27).
∜81 = 3, √0.49 = 0.7 and ∛(−27) = −3.
Check the decimal root: 0.7 × 0.7 = 0.49.
Pause and explainWhy is ∛(−8) negative?
Because (−2) × (−2) × (−2) = −8, so the cube root is −2.
Your turn. Recall, then calculate.
Start these questions by reading each symbol aloud: “squared”, “square root” or “cube root”. That short pause helps you avoid replacing a power with an ordinary multiplication.
Write the value of every power or root on its own line, then apply the remaining operations. For square roots, check the answer by squaring it; for cube roots, check by cubing it.
Write down √196, √225, ∛64 and ∛1000.
Reveal worked answer
Work out 4³ − 3² × √25.
Reveal worked answer
Calculate 2⁶ and ∜16.
Reveal worked answer
Find ∛(−125) and explain why √81 is not −9.