Properties of Exponents

Exponents

Multiplications are repeated addition. When multiplications are to be repeated, there are known as exponents.

22 = 2 x 2 means 2 has to be multiplied with itself.

Similarly,

84 = 8 x 8 x 8 x 8

am = a x a x a …upto m times.

Properties of exponents

  1. am*an = a(m+n)
  2. (abcd)n = an * bn * cn * dn
  3. (a⁄b)n = an⁄bn
  4. (-a)n = (-1)n * an = -an if n is odd andan if n is even
  5. (am)n = amn
  6. a-n = 1⁄an
  7. (a⁄b)-n = (b⁄a)n
  8. am⁄ an = am-n if m> n= 1⁄a(n-m) if n> m= 1 if m = n
  9. a0 = 1

Unit Digits

Number system is interesting. They form patterns.
For exponents too, numbers follow certain patterns. Here is the pattern followed by unit digits of the base number and the value of their exponents.
Based on the unit digit of the base numbers, the unit digits of their exponent results show a pattern.
for example: the unit digit of 2n follow the pattern as:
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256

The unit digits follow the pattern of 2, 4, 6, 8 in a cycle of 4.

Refer to the table below to see the pattern:

Unit digit of the base number Periodicity Unit digit of 24n+r for r =
0 1 2 3
1 1 1
2 4 6 2 4 8
3 4 1 3 9 7
4 2 6 4
5 1 5
6 1 6
7 4 1 7 9 3
8 4 6 8 4 2
9 2 1 9
0 1 0

How to use the table

If you have to find the unit digit in an, refer to the table above and based on the unit digit in number a, look into the corresponding row.

Now, look at the periodicity column. divide n by periodicity and find out remainder.

Now refer to the corresponding column. The unit digit of the exponent outcome corresponds to the number provided in the column.

for example if you have to find out the unit digit in 31999:

for 3, periodicity is 4, so we divide 1999 by 4. The remainder is 3.

For the row corresponding to 3 and for column corresponding to remainder 3, the unit digit is 7. Hence the unit digit in 31999 would be 7.

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