Topic/Skill Definition/Tips
Topic: Indices Example
1. Square The number you get when you multiply a 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,
Number number by itself. 144, 169, 196, 225…
9²=9 × 9=81
2. Square Root The number you multiply by itself to get √ 36=6
another number.
because 6 ×6=36
The reverse process of squaring a number.
3. Solutions to Equations involving squares have two Solve x 2=25
2
x =… . solutions, one positive and one negative.
x=5∨x=−5
This can also be written as x=± 5
4. Cube The number you get when you multiply a 1, 8, 27, 64, 125…
Number number by itself and itself again. 3
2 =2 ×2 ×2=8
5. Cube Root The number you multiply by itself and √3 125=5
itself again to get another number.
because 5 ×5 ×5=125
The reverse process of cubing a number.
6. Powers of… The powers of a number are that number The powers of 3 are:
raised to various powers.
1
3 =3
2
3 =9
3
3 =27
3 4=81 etc.
7. When multiplying with the same base 5
7 ×7 =7
3 8
Multiplication (number or letter), add the powers. a 12 ×a=a 13
Index Law 5 8
4 x ×2 x =8 x
13
a m × an=am+n
8. Division When dividing with the same base (number 7
15 ÷15 =15
4 3
Index Law or letter), subtract the powers. 9 2
x ÷ x =x
7
11 3 8
20 a ÷5 a =4 a
a m ÷ a n=a m−n
9. Brackets When raising a power to another power, ( y ¿¿ 2)5= y 10 ¿
Index Laws multiply the powers together. 4
(6¿ ¿3) =6 ¿
12
(5 x ¿¿ 6)3=125 x18 ¿
(a ¿¿ m)n=amn ¿
10. Notable p= p1 999990 =1
Powers 0
p =1
11. Negative A negative power performs the reciprocal. −2 1 1
3 = =
Powers −m 1 3 9
2
a = m
a
12. Fractional The denominator of a fractional power acts 2
2
Powers as a ‘root’. 27 3 =( √3 27 ) =3 2=9
( ) ( ) ()
3
The numerator of a fractional power acts as 25 2
=
3
√ 25 = 5 3= 125
a normal power. 16 √16 4 64
Mr A. Coleman Glyn School
Topic: Indices Example
1. Square The number you get when you multiply a 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,
Number number by itself. 144, 169, 196, 225…
9²=9 × 9=81
2. Square Root The number you multiply by itself to get √ 36=6
another number.
because 6 ×6=36
The reverse process of squaring a number.
3. Solutions to Equations involving squares have two Solve x 2=25
2
x =… . solutions, one positive and one negative.
x=5∨x=−5
This can also be written as x=± 5
4. Cube The number you get when you multiply a 1, 8, 27, 64, 125…
Number number by itself and itself again. 3
2 =2 ×2 ×2=8
5. Cube Root The number you multiply by itself and √3 125=5
itself again to get another number.
because 5 ×5 ×5=125
The reverse process of cubing a number.
6. Powers of… The powers of a number are that number The powers of 3 are:
raised to various powers.
1
3 =3
2
3 =9
3
3 =27
3 4=81 etc.
7. When multiplying with the same base 5
7 ×7 =7
3 8
Multiplication (number or letter), add the powers. a 12 ×a=a 13
Index Law 5 8
4 x ×2 x =8 x
13
a m × an=am+n
8. Division When dividing with the same base (number 7
15 ÷15 =15
4 3
Index Law or letter), subtract the powers. 9 2
x ÷ x =x
7
11 3 8
20 a ÷5 a =4 a
a m ÷ a n=a m−n
9. Brackets When raising a power to another power, ( y ¿¿ 2)5= y 10 ¿
Index Laws multiply the powers together. 4
(6¿ ¿3) =6 ¿
12
(5 x ¿¿ 6)3=125 x18 ¿
(a ¿¿ m)n=amn ¿
10. Notable p= p1 999990 =1
Powers 0
p =1
11. Negative A negative power performs the reciprocal. −2 1 1
3 = =
Powers −m 1 3 9
2
a = m
a
12. Fractional The denominator of a fractional power acts 2
2
Powers as a ‘root’. 27 3 =( √3 27 ) =3 2=9
( ) ( ) ()
3
The numerator of a fractional power acts as 25 2
=
3
√ 25 = 5 3= 125
a normal power. 16 √16 4 64
Mr A. Coleman Glyn School
