Overview
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In the realm of Mathematics, various laws of exponents exist. These rules are instrumental in solving numerous mathematical problems, particularly those involving recurrent multiplication operations. The laws of exponents aid in simplifying the processes of multiplication and division, thus making problem-solving easier. In this article, we will explore the six crucial laws of exponents, complemented by numerous solved examples.
Table of Contents:
Exponents represent the number of times a number is multiplied by itself. For instance, multiplying 9 by itself three times can be represented as 93. In this case, the exponent is ‘3’, which signifies the number of times 9 is multiplied. 9 is the base, which is the number being multiplied. In essence, exponents or powers denote the number of times a number should be multiplied by itself. If the power is 2, it means the base number is multiplied twice with itself. Here are some examples:
If a number ‘b’ is multiplied by itself n-times, it is represented as bn, where b is the base and n is the exponent.
Exponents obey certain rules, known as its laws. These laws aid in the simplification of expressions. Let's delve into the laws of exponents in more detail.
As we have discussed, there are several laws or rules for exponents. The essential laws of exponents are as follows:
Now, let's discuss each of these laws in detail, complete with examples.
According to this law, for any non-zero number a:
where m and n are real numbers.
Example 1: Simplify 44 × 42
Solution: 44 × 42 = 44+2 = 46
Example 2: Simplify (−3)-2 × (−3)-4
Solution: (−3)-2 × (−3)-4 = (-3)-2-4 = (-3)-6
According to this rule:
Here, a is a non-zero number and m and n are integers.
Example 3: Simplify 5-3 / 5-1
Solution: 5-3 / 5-1 = 5-3+1 = 5-2 = 1/25
According to this law, if ‘a’ is the base, then the power raised to the power of base ‘a’ gives the product of the powers raised to the base ‘a’, such as;
Here, a is a non-zero number and m and n are integers.
Example 4: Express 162 as a power with base 4.
Solution: We have, 4×4 = 16 = 42. Therefore, 162 = (42)2 = 44.
According to this rule, for two or more different bases, if the power is the same, then;
Here, a is a non-zero number and n is an integer.
Example 5: Simplify and write the exponential form of: 1/6 x 3-2
Solution: We can write, 1/6 = 2-2. Therefore, 2-2 x 3-2 = (2 x 3)-2 = 6-2.
According to this rule, the fraction of two different bases with the same power is represented as;
Here, a and b are non-zero numbers and n is an integer.
Example 6: Simplify the expression and find the value: 252 / 52
Solution: We can write the given expression as; (25/5)2 = 52 = 25
According to this rule, when the power of any integer is zero, then its value is equal to 1, such as;
a0 = 1
Here, ‘a’ is any non-zero number.
Example 7: What is the value of 70 + 32 + 60 + 81 - 41?
Solution: 70 + 32 + 60 + 81 - 41 = 1 + 9 + 1 + 8 - 4 = 15
According to this rule, if the exponent is negative, we can change the exponent into positive by writing the same value in the denominator and the numerator holds the value 1. The negative exponent rule is given as:
a-m = 1/am
Example 8: Find the value of 3-2.Solution:
Here, the exponent is a negative value (i.e., -2). Thus, 3-2 can be written as 1/32, which equals 1/9.
The fractional exponent rule is used if the exponent is in the fractional form. The fractional exponent rule is given by:
Here, a is called the base, and 1/n is the exponent, which is in the fractional form. Thus, a1/n is said to be the nth root of a.
Example 9: Simplify: 811/2
Solution:
Here, the exponent is in fractional form. (i.e., ½). According to the fractional exponent rule, 811/2 can be written as √81, which equals 9.
Simplify the following expressions using the laws of exponents:
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