An exponent tells you how many times a number (called the base) is multiplied by itself. Exponents are a way to show repeated multiplication of the same number.
Exponents act like a shortcut for repeated multiplication.
For example: 23 means 2 is multiplied by itself three times = 2×2×2 = 8
Exponent Examples
Exponents are mathematical symbols used to represent the multiplication of the same number multiple times. They help us express large values in a simpler form by indicating how many times a number is multiplied by itself.
Some examples are given below:
Example 1: If a file initially has a size of 50 MB and its size is halved in each step, what will be the size after 5 steps?
Solution:
The file size after 5 steps can be calculated using the formula:
Final Size = 50 x (1/2)5
After solving this, the final size will be approximately 1.5625
Example 2: Suppose we have a number 5 which is multiplied by itself 5 times, then this is expressed as, 5×5×5×5×5
As we can see, this is a very tedious way of representing the number in exponent; this is represented as,
5×5×5×5×5 = 55 = 3125
Thus, the exponent is a very short way of representing large numbers.
Negative Exponents
Negative Exponent is nothing but the exponents of the reciprocal numbers; thus, negative exponents are easily solved by taking the reciprocal and then easily solving the exponent using the normal rules. This is represented as,
x-n = (1/x)n
Suppose we have to solve for the negative exponent (2)-3 then,
(2)-3 = (1/2)3 = 1/8
Thus, exponent rules can be applied directly to simplify expressions involving exponents, including negative exponents.
Exponents with Fractions
The exponents with the fraction are also called the radicals. These are the exponents that have a fraction of their power. The square root, cube root, nth root, and others are all called exponents with fractions.
We represent the fraction exponents as,
- Square Root = √()
- Cube Root = 3√()
- nth Root = n√()
Now the fraction exponent is solved in two parts. In the first part, we solve the denominator and then solve the numerator, which is represented as,
xn/m = {(x)1/m}n
Here, we first solve (x)1/m and then take its nth power to get the final answer. This can be understood by the example added below,
Example: Simplify 43/2
Solution:
= 43/2
= (41/2)3
= 23 = 8
Decimal Exponents
Decimal Exponents are nothing but another way of representing the fraction exponents. If any exponent is given in the decimal form then we first change it into fraction form and then easily solve for the fraction form.
This can be understood by the example added below,
Example: Simplify 41.5
Solution:
= 41.5 (As, 1.5 = 3/2)
= 43/2
= (41/2)3
= 23 = 8
Exponent Table
| Type of Exponent | Expression | Expansion | Simplified value |
|---|---|---|---|
| Zero exponent | 60 | 1 | 1 |
| One exponent | 41 | 4 | 4 |
| Exponent and power | 23 | 2 × 2 × 2 | 8 |
| Negative exponent | 5-3 | 1/53 = 1/(5 × 5 × 5) | 1/125 |
| Rational exponent | 91/2 | √9 | 3 |
| Multiplication | 32 × 33 | 3(2 + 3) = 35 | 273 |
| Quotient | 75/ 73 | 7(5 – 3) = 72 | 49 |
Power of Product | 32 × 42 | ( 3 x 4) 2 = 122 | 144 |
| Power of an exponent | (82)2 | 8(2 × 2) = 84 | 4096 |
Scientific Notation with Exponents
Scientific Notation is a way of writing very large numbers as very small numbers. In scientific notation, the numbers are represented in the multiple of 10. The number is first converted into its unit form and then the number is multiplied with the power of 10 to get the number in scientific notation.
These numbers are useful in writing very large and very small numbers. Suppose we have to write 15670000, then in scientific notation it is represented as 1.567×107
Any number can be easily represented in the scientific notation by following the steps added below.
- Step 1: If the number is greater than onemark the decimal digit after the first digit from the starting of the number.
- Step 2: Then multiply the number with the 10 raise to the power as their are digits after the decimal or point (include zero in the counting)
- Step 3: If the number is smaller than one shift the decimal to the first digit counting from the left of the number excluding zeros.
- Step 4: Then multiply the number with the 10 raise to the negative power as their are digits from which the decimal is shift.
Example 1: Convert 134500000000 into scientific notation.
Solution:
= 134500000000
= 1.345 × 108
Example 2: Convert 0.0000001345 into scientific notation.
Solution:
= 0.0000001345
= 1.345 × 10-7
➢Practice: Solved Examples
