Exponentiation
Exponentiation is a process or operation of taking the exponent of a number. If x is an integer raised to n which is a positive integer, then it can be expressed as xn. This process is known as exponentiation in mathematics. Let us learn more about its rules and properties in this article.
| 1. | What is Exponentiation? |
| 2. | Exponentiation Formula |
| 3. | Exponentiation Properties |
| 4. | Binary Exponentiation |
| 5. | FAQs on Exponentiation |
What is Exponentiation?
Exponentiation is the process of doing repeated multiplication by writing the given expression in the form of ab, where a is the base and b is the exponent. Suppose we are multiplying 5 ten times, so instead of writing it as a multiplication fact, we use the exponentiation algorithm to write it as 510.
Exponentiation Formula
There is a specific way to write expressions or numbers in the form of exponentiation. The exponentiation formula with 'a' as the base and 'n' as the exponent is given below:
By using this formula of exponentiation, we can easily write any repeated multiplication fact in the exponential form.
Exponentiation Properties
There are some exponentiation properties that will help you to simplify expressions with powers easily. They allow us to rewrite exponents in a simplified manner to make it easy for us to do calculations. The list of exponentiation properties or the basic rules of exponentiation are given below:
| Exponentiation Property | Formula | Example |
|---|---|---|
| Zero Exponent | a0 = 1 | 20 = 1 |
| Identity Exponent | a1 = a | 31 = 3 |
| Product of Powers | am × an = am+n | 22 × 23 = 22+3 = 25 |
| Quotient Property | am/an = am-n | 55/53 = 55-3 = 52 |
| Negative Exponent Property | a-m = 1/am; (a/b)-m = (b/a)m | 7-2 = 1/72 |
| Power of a Power | (am)n = amn | (23)4 = 212 |
| Power of a Product | (ab)m = ambm | (2 × 3)6 = 26 × 36 |
| Power of a Quotient | (a/b)m = am/bm | (2/3)4 = 24/34 |
We can easily solve small exponents by using repeated multiplication. For example, to find the value of 23, we can multiply 2 three times. But, how to solve expressions with large exponents? We can simplify such numbers or expressions using binary exponentiation or exponentiation by squaring explained below.
Binary Exponentiation
Binary exponentiation, also known as exponentiation by squaring and square-and-multiply algorithm, is used to calculate the values of large exponents, say 4103. It is a trick that uses base-2 numbers to compute the value of expressions involving large exponents. In exponentiation by squaring, we use the following formulas depending on whether the exponent is even or odd:
For an instance, if the exponent is an odd number, let's say 4103, then we can write it as 4(42)51. Still, it is not possible to multiply 42 fifty-one times, so again we will apply the same formula. Now, we get 4[16(162)25]. But still, the new problem is of the same type, we can apply the same observation again. We will have to apply this formula again and again to reduce this expression which can make it more confusing and complicated. So, here is a trick for you. The repeated application of this algorithm is the same as decomposing the exponent (from base 10 to base 2 or from decimal to binary expansion) into a sequence of squares and products. So, instead of applying this formula, again and again, we can find the binary equivalent of the exponent. Let us solve 410.
410 (base 10 exponent) = 41010 (exponent is converted to its binary equivalent)
By expanding the exponent further, we get, 10102 = 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20. It can be simplified as 8 + 0 + 2 + 0.
410 = 4(8+0+2+0)
410 = 48 × 42
In this way, we have broken down a large exponent into parts. Now, 48 can be easily solved as 164, or 2562, and 42 is equal to 16. Instead of multiplying 4 ten times, now we only need to multiply 256 × 256 × 16. This is known as the binary exponentiation method. This method is useful when dealing with expressions with large exponents.
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FAQs on Exponentiation
What is Exponentiation in Math?
Exponentiation in math is defined as the operation used to represent repeated multiplication. For example, if 10 is multiplied three times, then it can be written as "10 raised to 3" which means 103. Here, 10 is the base, and 3 is the exponent.
Why is Exponentiation Used?
As we know that multiplication is a way to represent repeated addition. Similarly, exponentiation is used to represent repeated multiplication. Our calculations become easier by applying exponentiation properties.
What is the Symbol of Exponentiation?
The caret (^) symbol is used to represent exponentiation. Sometimes, an uppercase 'E' or a lowercase 'e' are also used to denote exponents.
How to Calculate Exponentiation?
For smaller exponents, we can multiply the base repeatedly by itself n times, where n represents the exponent. But for larger exponents, we use exponentiation by squaring or square-and-multiply algorithm to break down exponent into smaller parts.
What is Another Notation for Exponentiation?
Another notation for exponentiation is the logarithm function. Every exponential expression can be represented in the log function as if logbn = a ⇒ ba = n.