KS3 · Maths
Index/power notation
Writing ten 2s with a × between each one gets old fast. There's a shorthand — and one easy trap hiding inside it.
Maths · Number
Same digits, different meaning
Which meaning does each card have? Tap a card, then tap the column you think it belongs in.
Still to sort
5 × 5 × 5 (0)
three 5s multiplied together
Where the line is: The raised 3 counts how many 5s to multiply. It is not a number to multiply the 5 by.
3 × 3 × 3 × 3 × 3 (0)
five 3s multiplied together
Where the line is: 5³ and 3⁵ use the same two digits, but the base is the digit on the line and the exponent is the raised one, so swapping them changes the meaning.
5 × 3 (0)
one 5 and one 3, multiplied once
Where the line is: No raised number and no repeated base: this is ordinary multiplication of two different numbers.
A power is a base with a small raised exponent, like the 5 and the 3 in 5³. The exponent is read 'to the power of', and an exponent of 2 or 3 also has a short name: 'squared' (8² is eight squared) and 'cubed' (8³ is eight cubed). Each card below really means one of the three things at the top of the columns. Tap a card, tap its column, then read why.
Predict, then check
Look closely at which number wears the little 2.
What is 2 × 5²? Commit to an answer, then reveal.
Calculator: powers and roots
Problem
Use a calculator to find 137⁴ and 10⁴, then see what square roots give you.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
One small raised number that counts how many times to multiply
What you need to know
- In a power such as 2³, the base (2) is the number being multiplied and the exponent (3) says how many copies of it are multiplied together: 2 × 2 × 2.
- Read a power aloud as 'to the power of': 7⁴ is seven to the power of four, and means 7 × 7 × 7 × 7.
- An exponent of 2 can be read 'squared' and an exponent of 3 'cubed': 8² is eight squared (8 × 8) and 8³ is eight cubed (8 × 8 × 8).
- To evaluate a power, write it as a repeated product and multiply: 4³ = 4 × 4 × 4 = 64 and 10⁴ = 10 × 10 × 10 × 10 = 10 000.
- In a product of different bases, count each base separately, write the bases in ascending order and leave a power of one unwritten: 2 × 5², not 2¹ × 5².
The big picture
A power is repeated multiplication of one number, written as a base with a small raised exponent that counts how many copies of the base are multiplied together. So 5³ means 5 × 5 × 5 = 125, not 5 × 3. You can read powers aloud, expand them, evaluate them, simplify products of different bases into index form, and use a calculator for powers and square roots.
Key points
Worked example
Problem
Write 6 × 6 × 6 × 6 as a power, say it aloud, and find its value.
⚠ Watch out
Reading 5³ as 5 × 3 = 15. The 3 counts the 5s, so 5³ = 5 × 5 × 5 = 125. A close cousin is multiplying different bases together: 2 × 5² is 2 × 25 = 50, not 10².
Memory hook
Count the copies, don't multiply by the count: the little number tells you how many copies of the big number to multiply.
Check yourself
Cover the page. Write 9³ as a repeated product and find its value. Why isn't it 27? Then write 9 × 9 × 9 × 9 × 4 × 4 in index form.
Flashcards
(12)In 2³, which number is the base and which is the exponent?
How do you say 3⁶ aloud, and what does it mean?
What are the special names for exponents 2 and 3?
Does 5³ mean 5 × 3?
Are 5³ and 3⁵ the same?
What is 10⁴ as a repeated product, and what is its value?
What is 2 × 5²?
Two conventions when you simplify a product into index form?
How do you evaluate 137⁴ on a calculator?
Which calculator button helps when the base is 10, and what does the square button do?
What does 'evaluate' mean?
How are squaring and the square root linked, and when is a square root a whole number?
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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Keep me postedMore KS3 Maths topics
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- Algebraic notation and conventions
- Angle sum in a triangle
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- Area of a circle
- Area of a trapezium
- Area of a triangle
- Area of composite rectilinear shapes
- Arithmetic sequences and finding the nth term
- Associative, commutative and distributive laws
- Bisecting an angle
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