Exponents and Power

Here in this article we are going to discuss about exponents and power. Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a:

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Just as multiplication by a positive integer corresponds to repeated addition:

A power is an exponent to which a certain quantity is increased. The expression x a is therefore called as "x to the power." The power can be an integer, real number, or complex number. But, the power of an actual number to a non-integer power is not fundamentally itself a real number. For example, x12 is real only for x>=0.

Properties of exponents

The most important identity fulfilled by integer exponentiation is xm+n = xm. xn. This identity has the consequence

xm-n =`x^m/x^n`

• for x ≠ 0, and

(xm)n=xm.n

• Another basic identity is (x. y)n = xn. yn

Raised to the positive power value:

A+2 = A * A.

A+3 = A * A * A

While the above term is given as A squared (A2) and A+3 is known as x cubed.

Examples:

3+2 = 3 * 3

The answer is same 9 .This is known as positive term.

Positive power in terms of cube:

3+3 = 3 * 3* 3

The answer is same 27.

Raised to the negative power value:

The term xm-1 = (`x^m/x` ). While m =1, we get x0 = 1 .so the term as written as

`x^n/x^m`  = xn-m

In the special case when n and m are equal, so

1 = `x^n/x^n`  = xn-n = x0.

The number which raised to the power 1 is the number itself in it

The value of the any power 0 is 1.

Solved Examples

Below are the examples on exponents and power-

Example 1:

Solve equation on exponents : (a3)(a4)

Solution:

Terms of what those exponents mean. "To the 3rd" means "multiplying three copies" and "to the 4th" means "multiplying four copies". By the simplification method the factors are then multiplied. It is of the form

(a3)(a4) = (aaa) (aaaa)

= aaaaaaa

= a7

Example 2:

Solve equation on exponents: x3 x8

Solution:

Terms of what those exponents mean. "To the 3rd" means "multiplying three copies" and "to the 8th" means "multiplying eight copies". By the simplification method the factors are then multiplied. It is of the form

(x3)(x8)= (xxx)(xxxxxxxx)

= xxxxxxxxxxx

= x11

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Example 3:

Solve: `(ab^3)/a^2`

Solution:

Terms of what those exponents mean. "To the 3rd" means "multiplying three copies" and "to the 2th" means "multiplying two copies". By the simplification method the factors are then multiplied. It is of the form

`(ab^3)/a^2`  = b3a1-2

= `b^3/a`