Using Positive Exponents

Positive Exponents

Repeated multiplication can easily be represented by using exponential notation. The exponent is also called as power or index. When we say 3⁵, the base is 3 and the index is 5. The index tells us how many times the base must be multiplied. Here, the base 3 must be multiplied 5 times. That is

3⁵ = 3 × 3 × 3 × 3 × 3

In general, x^m indicates that x is repeatedly multiplied m times.

x^m = x • x • x •……. m times

All the Rules of Exponents apply to the Positive exponents. For any real number a and positive integers m and n

a) a^m × aⁿ = a^m+n

b) a^m ÷ aⁿ = a^m-n

c) (a^m)ⁿ = a^mn

d) (ab)^m = a^mb^m

e) (a/b)^m = a^m/b^m

Simplification of numbers with Positive Exponents

It is important to be able to simplify the arithmetic expressions with positive exponents.

(7³× 7²) / 7 = 7³⁺² / 7

= 7⁵/7

= 7^5-1

= 7⁴

Simplification of variables and Polynomials with Positive Exponents

The Rules of Exponents will work for the Variables and Polynomials as well! We can use the same rules effectively to simplify the polynomials. For example,

(x² × y³)⁴ = (x²)⁴(y³)⁴ Using (ab)^m = a^mb^m

= x⁸y¹² Using (a^m)ⁿ = a^mn

Sometimes we have to use the FOIL method to simplify the polynomial with exponents.

(x² + y³)² = (x² + y³) (x² + y³)

To simplify further we have to use FOIL method.

(x² + y³) (x² + y³) = x² • x² + y³ • x² + x² • y³ + y³ • y³

= x⁴ + 2x²y³ + y⁶ We have used, a^m • aⁿ = a^m+n

Note: (x² + y³)² ≠ (x²)²(y³)²