To understand simplification of exponents we first need to establish the rules of exponents.
Exponent rules:
1. b^m * b^n = b^(m+n)
2. b^m/b^n = b^(m-n)
3.(b^m)^n = b^mn
4.v(n&b^m ) = b^(m/n)
Exponentiation:
We know that multiplication corresponds to repeated addition. In the same way, exponentiation corresponds to repeated multiplication. In other words, exponentiation refers to the process of repeated multiplication. For example we can write, 4*4*4 as 4^3 or 5*5*5*5 = 5^4 etc. In general terms, b*b*b*b…. n times = b^n. Here, b is the base and n is called the exponent or the index.
b^2 is usually read as b squared. b^3 is read as b cubed; where as b^4 is read as ‘b raised to power 4’. In the same way b^(any other number) is read as ‘b raised to the power ______’.
Properties of exponents:
1. Exponent can be any real number.
2. When exponent is zero, the value of the term becomes equal to 1. That is to say that b^0 = 1
3. Exponent of one results in the base itself. So, b^1 = b.
4. ?(b?^(n)m) is not the same as b^(n^m ). ?(b?^(n)m) = b^mn where as b^(n^m ) = b^n^m.
5. When exponent is negative it is same as the positive exponent of the reciprocal of base. So, b^(-n) = (1/b)^n.
Rational exponents:
We saw above that exponent can be any real number. But for now we shall look at rational exponents only. A rational exponent would be of the type m/n. Therefore the number with rational exponent would look like this : b^(m/n). Based on the rules of exponents that we saw earlier, we can say that, b^(m/n) = v(n&b^m ). In other words it’s the nth root of b raised to power m. A number with a rational exponent may or may not itself be a rational number. For example, 4^(8/4) = 4^2 = 16. However, 3^(5/2) can be written as v(2&3^5 ) = v(3^4 * 3^1) = 3^(4/2) * 3^(1/2) = 3^2 * 3^(1/2) = 9 * v(3) = 9v(3) is an irrational number.
Solved examples:
1. Simplify: x^6 * x^5
Solution: x^6 * x^5 = x^(6+5) = x^11
2. Simplify: t^10/t^8
Solution: t^10/t^8 = t^(10-8) = t^2
3. Simplify: 5x^3/3x^5
Solution: 5x^3/3x^5 = (5/3)*(x^3/x^5) = (5/3) * (x^(3-5)) = (5/3) * x^(-2) = 5/3x^2
4. Simplify: (125x^2y^3z^2)^0
Solution: (125x^2y^3z^2)^0 = 1. That is because when exponent is zero, the term becomes = 1
