GCSE · Maths · AQA · Spec 8300 · Foundation
Calculate with roots and integer indices
That small raised number in 5² isn't a multiplier. It's a count, and that one idea unlocks powers and roots.
See a power
View
Explore
Switch views, then tap a highlighted part
3² = 3 × 3 = 9. Nine tiles make a perfect square, which is why 9 is called a square number.
Switch between the views. Watch what the index counts.
Watch it done: powers and roots together
Problem
Work out 2⁵ − √49 + ∛27
Predict, then check
You've found roots that come out whole. Does that always happen?
You're about to type √50 into a calculator. Will the answer be an integer (a whole number)?
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
The small number counts. The root counts backwards.
What you need to know
- When the index is a positive whole number, it counts how many copies of the number are multiplied together: 4³ = 4 × 4 × 4 = 64. Zero and negative whole numbers are integer indices too, and the pattern shows what they mean. Each time the index goes down by 1, you divide by 4 once more: 4² = 16, 4¹ = 4, 4⁰ = 1, 4⁻¹ = 1/4, 4⁻² = 1/16. So for any number a that isn't 0, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A negative index means 'one over', not a negative answer: 4⁻² is 1/16, not −16.
- A square number is a number multiplied by itself (like 36 = 6 × 6). A cube number is a number multiplied by itself three times (like 125 = 5 × 5 × 5).
- A square root asks which number multiplied by itself gives this one: √81 = 9 because 9 × 9 = 81. A cube root asks the same question with three copies: ∛8 = 2 because 2 × 2 × 2 = 8.
- The square root of a perfect square is always an integer. For any other number, a calculator gives a square root that isn't a whole number, such as √50 = 7.071…
- In a calculation, work out any brackets, then the powers and roots, before the adding, subtracting, multiplying or dividing.
The big picture
A positive index (the small raised number) counts how many copies of a number are multiplied together: 5² = 5 × 5 = 25 and 2³ = 2 × 2 × 2 = 8. For zero and negative indices, a⁰ = 1 and a⁻ⁿ = 1/aⁿ (a ≠ 0): 5⁰ = 1 and 5⁻² = 1/25. A number multiplied by itself makes a square number; multiplied by itself three times, it makes a cube number. A root runs this backwards: √49 asks which number times itself gives 49 (7), and ∛27 asks which number times itself three times gives 27 (3). A perfect square always has a whole-number square root. Any other number's square root can be found on a calculator, but it won't be a whole number.
Key points
Worked example
Problem
A square rug has an area of 144 cm². How long is each side of the rug?
⚠ Watch out
Reading the index as a times sign: 5² is not 5 × 2 = 10, and 2³ is not 2 × 3 = 6. The same slip shows up with roots: √16 is not 16 ÷ 2 = 8. It's 4, because 4 × 4 = 16.
Memory hook
When it's a positive whole number, the little number is a headcount, not a times sign. 5² is two 5s multiplied together: 5 × 5 = 25. A root asks it backwards: which number, multiplied by itself, made this one?
Check yourself
Is 200 a perfect square? Decide without a calculator by multiplying whole numbers by themselves. Then say what that tells you about √200.
Flashcards
(11)What does the index 5 in 10⁵ tell you to do?
What's the difference between a square number and a cube number?
Say the square numbers from 1 × 1 to 10 × 10.
Say the first five cube numbers.
What question does √81 ask?
What question does ∛1000 ask?
How do you check that √196 = 14?
When is a square root guaranteed to be an integer?
Your calculator shows √30 = 5.477… What does that tell you about 30?
Are 2⁵ and 5² the same?
What are 7⁰ and 10⁻²?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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