The exponent of a number shows how many times a number is multiplied by itself. For example, 3 4 means 3 is multiplied four times by itself, that is, 3 × 3 × 3 × 3 = 3 4 , and here 4 is the exponent of 3. Exponent is also known as the power of a number and in this case, it is read as 3 to the power of 4. Exponents can be whole numbers, fractions, negative numbers, or decimals. Let us learn more about the meaning of exponents along with exponents examples in this article.
| 1. | What are Exponents? |
| 2. | Laws (Properties or Rules) of Exponents |
| 3. | Negative Exponents |
| 4. | Exponents with Fractions |
| 5. | Decimal Exponents |
| 6. | Scientific Notation with Exponents |
| 7. | FAQs on Exponents |
The exponent of a number shows how many times the number is multiplied by itself. For example, 2 × 2 × 2 × 2 can be written as 2 4 , as 2 is multiplied by itself 4 times. Here, 2 is called the 'base' and 4 is called the 'exponent' or 'power'.
Exponent is the way in which large numbers are expressed in terms of powers. For example, 4 multiplied 3 times by itself can be expressed as 4 × 4 × 4 = 4 3 , where 3 is the exponent of 4. Observe the following figure to see how we express the exponent of a number. It shows that x n means that x is multiplied by itself 'n' times.
Here, in the term x n ,
Some examples of exponents are as follows:
Exponents are important because when a number is multiplied by itself many times, it is easy to express it in the form of exponents. For example, it is easier to write 5 7 rather than writing it as 5 × 5 × 5 × 5 × 5 × 5 × 5.
The properties of exponents that are also known as the laws of exponents are used to solve problems involving exponents. These properties are also considered as major exponents rules. The basic properties of exponents are given below.
A negative exponent tells us how many times we have need to multiply the reciprocal of the base. For example, if it is given that a -n , it can be expanded as 1/a n . It means we have to multiply the reciprocal of a, i.e., 1/a 'n' times. Negative exponents are used while writing fractions with exponents. Some examples of negative exponents are 2 × 3 -9 , 7 -3 , 67 -5 , etc. We can convert these into positive exponents as follows:
If the exponent of a number is a fraction, it is known as a fractional exponent. Square roots, cube roots, n th root are parts of fractional exponents. A number with power 1/2 is termed as the square root of the base. Similarly, a number with a power of 1/3 is called the cube root of the base. Some examples of exponents with fractions are 5 2/3 , -8 1/3 , 10 5/6 , etc. We can write these as follows:
If the exponent of a number is given in the decimal form, it is known as a decimal exponent. It is slightly difficult to evaluate the correct answer of any decimal exponent so we find the approximate answer for such cases. Decimal exponents can be solved by first converting the decimal into fraction form. For example, 4 1.5 can be written as 4 3/2 which can be simplified further to get the final answer 8, i.e., 4 3/2 = (2 2 ) 3/2 = 2 3 = 8.
Scientific notation is the standard form of writing very large numbers or very small numbers. In this, numbers are written with the help of decimals and powers of 10. A number is said to be written in scientific notation when a number between 0 to 10 is multiplied by a power of 10. In the case of a number greater than 1, the power of 10 will be a positive exponent, while in the case of numbers less than 1, the power of 10 will be negative. Let us understand the steps for writing numbers in scientific notation with exponents:
By following these two simple steps we can write any number in the standard form with exponents, for example, 560000 = 5.6 × 10 5 , 0.00736567 = 7.36567 × 10 -3 .
To learn more about the use of exponents in writing scientific notation of numbers, visit the following articles:
Tips and Tricks:
☛ Related Topics on Exponents
Check a few more interesting articles based on the exponents in math.
Example 1: Find the product of the following expressions: a 5 × b 3 × a 8 Solution: Let us find the product of a 5 × b 3 × a 8 using the exponents rule = a m × a n = a (m+n) This will be a 5 × b 3 × a 8 = a 5+8 × b 3 = a 13 × b 3 = a 13 b 3
Example 2: Find the product of 5 7 × 5 3 using the properties of exponents. Solution: 5 3 × 5 7 = 5 10 (using exponents formula = a m × a n = a (m+n) )
Example 3: Simplify the following expression: p 12 ÷ p 4 q. Solution: The given expression is p 12 ÷ p 4 q. To simplify this expression, we use the law of quotient of exponents which says a m /a n = a m-n . ⇒ p 12 /p 4 q ⇒ p 12-4 /q ⇒ p 8 /q Therefore, p 12 ÷ p 4 q = p 8 /q
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