Algebra - Exponent
- What is an Exponent?
- Examples of Exponent
- Reading Exponents
- Laws of Exponents
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is an Exponent?
An exponent of a number tells how many times that number should be multiplied by itself. For example: `4^3` means `4` is multiplied by itself `3` times. The number `4` here is called base which means the number that is to be multiplied and `3` is called exponent which means the number of times we multiply the base by itself. Like the base, the exponent can take any value be it integer, fraction, or decimal. Base and exponent can be both positive or negative.
Examples of Exponent
Below are some examples of exponents:
- `6^2= 6 × 6`
- `4^5= 4 × 4 × 4 × 4 × 4`
- `13^2= 13 × 13`
- `(-5)^6=(-5) × (-5) × (-5) × (-5) × (-5) × (-5)`
- `(8/7)^3 = (8/7) × (8/7) × (8/7) `
- `(0.5)^4 = (0.5) × (0.5) × (0.5) × (0.5)`
Reading Exponents
There are several ways to read `a^n`.Some of them are:
- `a` raised to the power of `n`.
- `a` raised to `n`.
- `a` to the `n^th` power.
- `a` to the power `n`.
When the exponent is `2` , for example, `n^2` is called n square or if the exponent is `3` then `n^3` is called `n` cube.
Laws of Exponents
There are some rules that help in solving problems related to exponents. Some of them are mentioned below:
`a`. Product Law: If two numbers having the same base are multiplied, then we add the exponents of the two numbers.
`m^n × m^p = m^(n+p)`
`b`. Quotient Law: If two numbers having the same base are divided, then we subtract the exponents of the two numbers.
`m^n/m^p = m^(n-p)`
`c`. Power Law: If a number with an exponent is raised to another exponent, then we multiply the exponents.
`(m^n)^p=m^np`
`d`. Negative Law: If a number has a negative number as an exponent, the exponent can be converted into a positive number by taking its reciprocal.
`m^(-n) = 1/m^n`
`e`. Zero Exponent Law: If a number is raised to 0, then its value is equal to 1.
`m^0 = 1`
`f`. Power Product Law: If two powers with different bases having the same exponent are multiplied, then the result is the multiplication of bases raised to the common exponent.
`m^a × b^a = (mb)^a`
`g`. Power Quotient Law: If two powers with different bases having the same exponent are divided, then the result is the quotient of bases raised to the common exponent.
`m^n/b^n =(m/b)^n`
Solved Examples
Example `1`: Find the value of `8^4`.
Solution:
`8^4 = 8 × 8 × 8 × 8` = `4096`
Example `2`: Simplify the given expression :`3^2 × 3^3 × 3^2 × 3^7`
Solution:
`3^2 × 3^3 × 3^2 × 3^7 = 3^(2+3+2+7) = 3^14=4782969`
Example `3`: Simplify: `{(3/4)^2}^4`
Solution:
`{(3/4)^2}^4=(3/4)^(2⨰4) = (3/4)^8 = 6561/65536`
Example `4`: Solve the following:
`3^4 × 4^4`
Solution:
`3^4 × 4^4 = (3 × 4)^4 = 12^4 = 20736`
Example `5`: Solve the following:
`12^7 ÷ 12^8`
Solution: `12^7 12^8 =(12)^(7-8) = (12)^(-1) = 1/12`
Practice Problems
Q`1`. Find the value of `3^3 × 3^4 × 9^5`.
- \(3^{17}\)
- \(3^{14}\)
- \(3^{21}\)
- \(3^{10}\)
Answer: a
Q`2`. What is the value of `(1992)^0` ?
- `0`
- `10`
- `1992`
- `1`
Answer: d
Q`3`. Find the value of `5^6/5^8`.
- `25`
- `1/25`
- `5^(14)`
- `1/(5^(14))`
Answer: b
Q`4`. Simplify `(16/4)^(-2)`.
- `8`
- `1/(4096)`
- `4096`
- `1/16`
Answer: d
Frequently Asked Questions
Q`1`. What is an exponent?
Answer: An exponent, or power, is a mathematical notation that represents the number of times a base is multiplied by itself. In the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent.
Q`2`. How do you read expressions with exponents?
Answer: The expression \(a^n\) is read as "`a` raised to the power of `n`" or simply "`a` to the power of `n`."
Q`3`. What is the result of a number raised to the power of zero?
Answer: Any non-zero number raised to the power of zero is equal to `1`. It is a fundamental exponent property.
Q`4`. How do you multiply two expressions with the same base but different exponents?
Answer: When multiplying two expressions with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m + n}\).