Indices and Surd Rules

Question 1

What is a^0 the same as? (‘a’ can be any number)

Answer 1

It is always 1 or -1, depending on if ‘a’ is positive or negative.

Question 2

What is a^-n the same as? (‘a’ and ‘n’ can be any number)

Answer 2

It is 1/a^n or -1/a^n, depending on if ‘a’ is positive or negative.

Question 3

What is a^1/n the same as? (‘a’ and ‘n’ can be any number)

Answer 3

It is nVa (‘V’ is a root, so nVa is the same as;

n’th root out of ‘a’)

Question 4

What is a^m/n the same as? ( ‘a’, ‘n’ and ‘m’ can be any number)

Answer 4

It is (nVa)^m ( 'V' is a root, so (nVa)^m is the same as; n'th root of a to the power of m)
This only applies if m>1.

Question 5

Name the 1st law of indices.

Answer 5

1st law: a^n x a^m = a^n+m

Question 6

Name the 2nd law of indices.

Answer 6

2nd law: a^n / a^m = a^n-m

Question 7

Name the 3rd law of indices.

Answer 7

3rd law: (a^n)^m = a^nm

Question 8

Example question: simplify 7^4 x 7^3

Answer 8

= 7^7

Question 9

Example question: simplify z^4 y^5 q^8 x z^-2 y^3 q^-10

Answer 9

= z^2 y^8 q^-2

Question 10

Example question: what is 12x8x18 as a product of prime numbers?

Answer 10

=2x2x3x2x2x2x3x3

=2^5x3^3

Question 11

Example question: what is 4^3x9^3x20^2 as a product of prime numbers?

Answer 11

=2^10x3^6x5^2

Question 12

Example question: what is (4mn^2)^3 / 2m^2n ?

Answer 12

=32mn^5

Question 13

Example question: what is 4y^4 x (3y)^2 ?

Answer 13

=36y^6

Question 14

Example question: what is p^7 / p^4 ?

Answer 14

=p^3

Question 15

Example question: what is 7^-3

Answer 15

=1/7^3

=1/343

Question 16

Example question: what is 64^1/3

Answer 16

=3V64 (‘3V’ means cube root)

= 4

Question 17

Example question: what is 27^4/3

Answer 17

=(3V27)^4

=81

Question 18

Example question: 125^-4/3

Answer 18

=1/(3V125)^4
=1/5^4
=1/625

Question 19

What can Va x Vb be simplified to? (‘V’ being any root)

Answer 19

= Vab ( ‘a’ and ‘b’ being multiplied and under the same root)

Question 20

What can Va / Vb be simplified to? (‘V’ being any root)

Answer 20

= Va/b ( ‘a’ being divided by ‘b’ and under the same root)

Question 21

What can mVnVa be simplified to? ( mVnVa meaning the root out of the root of ‘a’, mV and nV being different roots)

Answer 21

= mnVa (‘m’ and ‘n’ multiply to form one root)

Question 22

Example question: what is V2xV12.5 ?

Answer 22

=V(2x12.5)
=V25
=5

Question 23

Example question: what is 3V4 x 3V16 ? (‘3V’ meaning cube root)

Answer 23

=3V(4x16)
=3V64
=4

Question 24

Example question: what is V162 / V2 ?

Answer 24

= V192/2
=V81
=9

Question 25

Exampe question: what is 5V243 / 5V32 ? (‘5V meaning 5th root)

Answer 25

=5V(243/32)

=3/2