The readers may find that literatures on mathematics provide various definitions of powers. Some authors do not distinguish the term power from index/exponent. Throughout this article, a power is a mathematical expression in the form of br where b, \: r \in \mathbb{R}. The number r is called an index or exponent, and b is called a base. By the definition, br is a power with the base b and exponent r.

 

Case 1: r  is a natural number

Below are how br are defined for r = 1, 2, 3, …

b1 = b   ; b \in \mathbb{R}

br = b1 ⋅ b2 ⋅ b3 ⋅ … ⋅ br with b1 = b2 = b3 = … = br = b   ; r = 2, 3, 4, …

 

Case 2: r  = 0

b0 = 1   ; b ≠ 0

Note: 00 is not defined!

 

Example 1

53 = 5⋅5⋅5 = 125

(0.1)4 = 0.1 ⋅ 0.1 ⋅ 0.1 ⋅ 0.1 = 0.0001

 

Case 3: r  is a negative integer

b^{-n}=\frac{1}{b^n} ; n \in \mathbb{N}, b ≠ 0.

Note: \mathbb{N} is the set of all natural numbers.

 

Example 2

7^{-1}=\frac{1}{7^1}=\frac{1}{7}

3^{-4}=\frac{1}{3^4}=\frac{1}{3 \cdot 3 \cdot 3 \cdot 3} = \frac{1}{81}

 

Case 4: r  is a rational number

b^{\frac{1}{2}}=\sqrt{b} ; b \in \mathbb{R}, b ≥ 0

Thus, b½ is the square root of b.

 

Example 3

16^{\frac{1}{2}}=\sqrt{16}=4

 

b^{\frac{1}{2n}}=\sqrt[2n]{b} ; n \in \mathbb{N}, b ≥ 0

 

Example 4

10000^{\frac{1}{4}}=\sqrt[4]{10000}

 

b^{\frac{1}{2n+1}}=\sqrt[2n+1]{b} ; n \in \mathbb{N}

 

Example 5

125^{\frac{1}{3}}=\sqrt[3]{125}=5

(-32)^{\frac{1}{5}}=\sqrt[5]{-32}=-2

(-128)^{\frac{1}{7}}=\sqrt[7]{-128}=-2

 

b^{\frac{m}{2n}}=(b^{\frac{1}{2n}})^m=(b^m)^{\frac{1}{2n}} ; m, n \in \mathbb{N}, b ≥ 0.

 

Example 6

81^{\frac{3}{4}}=(81^{\frac{1}{4}})^3=(\sqrt[4]{81})^3=3^3= 27

4^{\frac{5}{2}}=(4^5)^{\frac{1}{2}}=1024^{\frac{1}{2}}=\sqrt{1024 }=32

 

b^{\frac{m}{2n+1}}=(b^{\frac{1}{2n+1}})^m=(b^m)^{\frac{1}{2n+1}} ; m, n \in \mathbb{N}

 

Example 7

(-125)^{\frac{5}{3}}={\left[ (-125)^{\frac{1}{3}} \right]}^5=(\sqrt[3 ]{-125})^5=(-5)^5=-3125

0.00001^{\frac{7}{5}}=(0.00001^{\frac{1}{5}})^7=(\sqrt[5]{0.00001})^ 7=0.1^7=0.0000001

8^{\frac{2}{3}}=(8^2)^{\frac{1}{3}}=64^{\frac{1}{3}}=\sqrt[3 ]{64}=4

 

b^{\frac{-m}{2n}}=\frac{1}{b^{\frac{m}{2n}}} ; m, n \in \mathbb{N}, b > 0

 

Example 8

81^{\frac{-3}{4}}=\frac{1}{81^{\frac{3}{4}}}=\frac{1}{(81^{\frac{ 1}{4}})^3}=\frac{1}{(\sqrt[4]{81})^3}=\frac{1}{3^3}=\frac{1}{27}

4^{\frac{-5}{2}}=\frac{1}{4^{\frac{5}{2}}}=\frac{1}{(4^5)^{ \frac{1}{2}}}=\frac{1}{1024^{\frac{1}{2}}}=\frac{1}{\sqrt{1024}}=\frac{1}{ 32}

 

b^{\frac{-m}{2n+1}}=\frac{1}{b^{\frac{m}{2n+1}}} ; m, n \in \mathbb{N}, b ≠ 0

 

Example 9

(-125)^{\frac{-5}{3}}=\frac{1}{(-125)^{\frac{5}{3}}}=\frac{1}{{ \left[ (-125)^{\frac{1}{3}}\right]}^5}=\frac{1}{(\sqrt[3]{-125})^5}=\frac{ 1}{(-5)^5}=\frac{1}{-3125}

8^{\frac{-2}{3}}=\frac{1}{8^{\frac{2}{3}}}=\frac{1}{(8^2)^{ \frac{1}{3}}}=\frac{1}{64^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{64}}=\frac{ 1}{4}

 

How if the exponents are irrational numbers? For example, how to compute 5^{\sqrt{3}}? Indeed, the number is irrational as well, whose decimal representation is an approximation. This will be discussed later on this website.

 

Some formulae related to powers

(Apply the following formulae cautiously, look at the description below R5.)

Let a, b, m, n \in \mathbb{R}. Then,

R1: b^m \cdot b^n = b^{m+n}

R2: \frac{b^m}{b^n}=b^{m-n}   ; b ≠ 0

R3: (b^m)^n=(b^n)^m=b^{mn}

R4: (a \cdot b)^n=a^n \cdot b^n

R5: (\frac{a}{b})^n=\frac{a^n}{b^n}   ; b ≠ 0

Cautions

The formulae R1 through R5 hold, provided that the powers and algebraic operations are defined. For example, we must not apply R1 in 05⋅⋅0-2 = 03 because 0-2 is undefined. As another example of misusing R1 is (-81)^{\frac{1}{4}} \cdot (-81)^{\frac{1}{2}}=(- 81)^{\frac{3}{4}}. It is incorrect because (-81)^{\frac{1}{4}} is undefined and so are (-81)^{\frac{1}{2}} and (-81)^{\frac{3}{4}}.

 

Exponential functions

The exponential function f is the real-valued function whose domain is \mathbb{R} and for every x ∈ \mathbb{R} f(x) = ex, where e is the Euler number. Below is the graph of the exponential function.

 

The function has the following properties.

  1. It is strictly increasing, that is x1 < x2 ⇒ f(x1) < f(x2).
  2. The coordinates of the y-intercept is (0,1).
  3. The x-axis is the asymptote to the curve of the function. The graph of the function gets closer and closer to the axis as x gets smaller and smaller.
  4. The formulae R1 through R3 above apply.

On

Leave a Reply

Your email address will not be published. Required fields are marked *

Related Topics