Key ideas
Powers
The power tells you how many times to multiply the number by itself.
25 = 2 × 2 × 2 × 2 × 2 = 32
25 is not 2 × 5.
Negative numbers
−32 = −3 × −3 = 9
−23 = −2 × −2 × −2 = −8
An even power gives a positive answer. An odd power keeps the negative.
Powers of fractions
Apply the power to the top and the bottom.
232 = 2232 = 49
Roots
A square root undoes squaring: 49 = 7, because 72 = 49.
A cube root undoes cubing: 64 = 4, because 43 = 64.
Learn these: square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and cube numbers 1, 8, 27, 64, 125, 1000.
Powers of whole numbers
Evaluate 34.(1 mark)
Write it out and multiply.34 = 3 × 3 × 3 × 3 four 3s= 9 × 3 × 3 = 27 × 3 = 81✓1
In plain EnglishThe power 4 means four 3s multiplied together. Multiply two at a time.
⚠ A common mistake: 34 is not 3 × 4 = 12.
Negative numbers and powers
Evaluate (a) −42 (b) −25.(2 marks)
Even power.−42 = −4 × −4 = 16✓1
In plain EnglishNegative times negative is positive.
Odd power.−25 = −2 × −2 × −2 × −2 × −2= 4 × 4 × −2 = −32✓2
In plain EnglishPair up the negatives: each pair makes a positive. With five, one negative is left over, so the answer is negative.
Powers of fractions
Evaluate 352.(1 mark)
Square the top and the bottom.352 = 3 × 35 × 5 = 925✓1
In plain EnglishThe power applies to the whole fraction, so square the numerator and the denominator.
⚠ A common mistake: squaring only the top, giving 95.
Evaluate 1122. Give your answer as a mixed number.(2 marks)
Make it top-heavy.112 = 32✓1
In plain EnglishYou can't square a mixed number in parts. Change it to an improper fraction first.
Square, then change back.322 = 94 = 214✓2
In plain English3 squared is 9 and 2 squared is 4. 9 quarters is 2 wholes and 1 quarter.
⚠ A common mistake: 1² + 122 = 114. That is not the same.
Square roots
Evaluate (a) 81 (b) 1649.(2 marks)
Which number squared gives 81?9 × 9 = 81, so 81 = 9✓1
In plain EnglishThink: what times itself gives 81?
Root the top and the bottom.1649 = 1649 = 47✓2
In plain EnglishFor a fraction, square root the numerator and the denominator separately.
Cube roots
Evaluate 125.(1 mark)
Which number cubed gives 125?5 × 5 × 5 = 125125 = 5✓1
In plain EnglishA cube root asks: what number, multiplied by itself three times, gives 125?
Powers and roots together
Evaluate 23 + 36 × 3.(2 marks)
Powers and roots first.23 = 8, 36 = 6✓1
In plain EnglishIn BODMAS, powers and roots come before multiplying and adding.
Then multiply, then add.8 + 6 × 3 = 8 + 18 = 26✓2
In plain EnglishMultiply before you add.
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Your turn
Non-calculator. Type fractions with a slash, like 4/9, and mixed numbers with a space, like 2 1/4. Negative answers need a minus sign. Stuck? Tap Need a hint?