Laws of Exponents for Real Numbers

Let ‘a’ be a positive number. Exponential notation is ax , where x is an integer. When x is a rational number, say p/q with p, an integer and q, a positive integer, we define ax by ax  = root(q)(a) p

Example:

53/8 = root(3)(5)8

laws of exponents for real numbers: Exponents Notation

When x is an irrational number, ax can be defined to represent a real number.

For any a > 0, ax can be defined and that it represents an unique real number u and write u = ax.

The real number u is written in the exponential form or in the exponential notation.

Here the positive number 'a' is called the base and x, the index or the power or the exponent.

The laws of indices which we have stated for integer exponents can be obtained for all real exponents.


laws of Exponents for Real Numbers

We state them here and call them, the laws of exponents for real number:

root(x)(root(y)(a))  = root(xy)(a)


root(x)(a / b)  =  root(x)(a) / root(x)(b)


(root(x)(a) )x = a


axxx ay = ax+y


ax// ay = ax-y


(ax)y = axy


a-x  = 1/ ax


axxx bx = abx


a0 = 1  for a != 0


(a/b)x = ax/bx


a1/x =  root(x)(a)


(root(x)(a) )y/x = (root(x)(a) )


root(x)(a) root(x)(b) = root(x)(ab)



Example sums for Exponents for Real Numbers

Example 1:

Find the exponential form of (25)1/17

Solution of example 1:

Given = (25)1/17

Exponential form  =  root(17)(25)

Example 2:

Simplify 25225 / 53253

Solution of example 2:

Given 25225 / 53253

Now we use the exponents formula for real numbers

By using

(i)axxx ay = ax+y

(ii)axxx bx = abx

We get

= 253 / (5(25))3

= 253 / 1253

= 253 / (5)3(25)3

Cancel the same variables

Now, we get

= 1 / 53

= 1 / 125

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Example 3:

Solve ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Solution of example 3:

Given = ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Now we use the exponents formula for real numbers

By using below laws of exponents formula we can simplify the given data.

(i) root(x)(root(y)(a)) = root(xy)(a)

(ii)axxx bx = abx

(iii)(ax)y = axy

(iv)(a/b)x = ax/bx

= ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Use the (i) formula

We get

= root(6)(49))92xx 62) / ((43)2+(5/25)2)

Use the (ii) formula

We get

= ((root(6)(64))542) / ((43)2+(5/25)2)

Use the (iii) formula

We get

= ((root(6)(64))542) / ((46)+(5/25)2)

Use the (iii) formula

We get

= ((root(6)(64))542) / ((46)+(52/252))

= (2 (542) / ((256) + (25 / 252))

Cancel the same elements

We get

= 2(542) / ((256) + (1 / 25))

= 2(542) / ((6400 + 1) / 25 )

= 2(2916) / (6401/25)

= 2(2916)(25) / (6401)

= 50(2916) / 6401

= 145800 / 6401

= 22.77