What is the Power in Mathematics?
The term “power” in mathematics refers to the process of increasing a base number to its exponent. “Base number” and “exponent” are the two fundamental elements of powers. The exponent represents the number of times the base number is multiplied, while the base number is defined as a number that’s being multiplied by itself. In simple terms, power is a number that is expressed with exponents. It denotes the multiplication of the same factor multiple times.
What is an Exponent?
Definition: The exponent of a number shows how many times we multiply the number itself. For example, \(2^3\) indicates that we multiply 2 by 3 times. Its extended form is written as 2 × 2 × 2. Exponent is also known as numerical power. It could be a whole number, a fraction, a negative number, or decimals.
What are Negative Exponents?
Definition: A negative exponent is used when 1 is divided by repeated multiplication of a factor. Say, \(\frac{1}{a}\) is given by \(a^{-1}\), where -1 is the exponent. If a number is raised to a negative exponent then it represents the reciprocal of it. For example, 5 raised to -3 is represented as \(5^{-3}\), which can also be written as \(\frac{1}{5}^3\).
Exponent Symbol
For representing the exponent of the base number, we use the symbol ^. This symbol (^) is called a carrot. For example, 3 raised in 2 can be written 3 ^ 2. Thus, 3 ^ 2 = 3 × 3, which is 9. The table below shows the representations of a few number sentences using objects.
Expression | Exponent representation |
|---|---|
| \(5 \times 5 \times 5\) | \(5^3,\) Base = 5, Exponent = 3 |
| \(3 \times 3 \times 3 \times 3 \times 3 \times 3\) | \(3^6,\) Base = 3, Exponent = 6 |
| \(8 \times 8 \times 8 \times 8\) | \(8^4,\) Base =8, Exponent = 4 |
Exponents Properties
Exponents’ properties or rules of exponents are used to solve problems involving exponents (exponents worksheets, 8th grade exponents worksheets). These structures are also considered to be the rules of the main exponents which must be followed when resolving the exponents. The characteristics of the exponents are listed in the table.
Name of the law | The Exponent law | Example | The Simplified Value |
|---|---|---|---|
Law Of Product | \(x^a\times\ x^b=x^{a+b}\) | \(2^3\times2^5=2^{3+5}={2^8}\) | 256 |
Law of Power of a Power | \(\left(x^a\right)^b=x^{ab}\) where x is a non-zero term and a and b are integers. | \(\left(2^2\right)^3=2^{2\times3}=2^6\) | 64 |
Product to a Power | \((xy)^a=x^ay^a\) | \((1\times3)^3=1^3\ \times3^3\) | 27 |
Law of Quotient | \(\frac{x^a}{x^b}=x^{a-b}\) where \(x\neq0\) | \(\frac{3^4}{3^2}=3^{4-2}=3^2\) | 9 |
Law of Zero Exponent | \(x^0=1\) | \(2^0=1\) | 1 |
Law of Negative Exponent | \(x^{-a}=\frac{1}{x^a}\) | \(2^{-3}=\frac{1}{2^3}\) | \(\frac{1}{8}\) |
Law of Power of a Quotient | \((\frac{x}{y})\ ^a=\frac{x^a}{y^a}\) | \((\frac{2}{3})\ ^5=\frac{2^5}{3^5}\) | \(\frac{32}{243}\) |
Solved Exponent Examples
Question 1:
In the garden, each tree has \(5^7\) leaves and there are about \(5^3\) trees in the garden. Calculate the total number of leaves in terms of solving the exponents.
Solution:
Number of trees in the garden = \(5^3\) and
number of leaves per tree = \(5^7\) .
The total number of leaves is: \(5 ^ 3 × 5 ^ 7 = 5 ^ {10}\) (using exponents formula = \(a^m \times a^n=a^{(m+n)}\)).
Therefore, the total number of leaves is \(5^{10}\).
Question 2:
10 books and 10 sheets of paper are placed in a stack. Find the total thickness of the stack if each book has a thickness of \(10^3\) mm and each sheet has a thickness of 0.01 mm.
Solution:
Thickness of 1 book = \(10^3\)mm
And,
Thickness of one paper = 0.01 mm
So, thickness of 10 books = \(10^3\)×10 = \(10^4\)mm ( using exponents formula : \(a^m\times a^n = a^{m+n})\)
And,
Thickness of 10 sheets of paper = \(0.01\times 10\) = 0.1 mm
Now, the total thickness of the stack is:
= \(10^4\)+0.1 mm
= \(10^4\)+\(10^{-1}\)mm (0.1=\(10^{-1}\))
Question 3:
The annual profit(in thousands of dollars) earned by a technology company a few years after opening is represented by the equation P = \(0.1{x^3}+3\) . How much more profit is earned in year 5 than in year 4?
Solution:
Use the equation to find the profits earned in year 4 and year 5. Then subtract the profit in year 4 from the profit in year 5 to determine how much more profit is earned in year 5.
profit earn in year 4 is
P = \(\ 0.1{\ x^3}+3\)
P = \(\ 0.1{\ 4^3}^+3\)
P = \(\ 0.1(64)+3\)
P = 6.4+3
P = 9.4
Profit earned in year 5 is,
P = \(0.1{\ x^3}+3\)
P = \(0.1{\ 5^3}+3\)
P = \(0.1(125)+3\)
P = 12.5+3
P = 15.5
So, the company earns 15.5 – 9.4 = 6.1 , or 6100 dollars more profit in year 5 than in year 4.
Power is an expression that represents the multiplication of a whole number. Generally, an is a power where a is the base and n is an exponent.
For example,
\(6^3\) is the power that shows 6 is multiplied by itself three times.
If the value of the base number is one, the base value remains the same.
For example,
\(5^1=5\). If the exponent is 0, then we get 1 as the end result.
For example,
\(5^0=1\)
Exponents have a variety of applications. Applications for a few exponents are listed below:
1)Exponents are widely used in computer games, scales, etc.
2)Scientific scales such as pH scale or Richter scale are based on exponents.
3)Used when calculating area, volume, and measurement problems.
4)They are widely used in the fields of science, engineering, economics, accounting, and finance.
5)Often used to represent the memory of a computer or laptop.