Negative exponents tell us that the power of a number is negative and it applies to the reciprocal of the number. We know that an exponent refers to the number of times a number is multiplied by itself. For example, 32 = 3 × 3. In the case of positive exponents, we easily multiply the number (base) by itself, but what happens when we have negative numbers as exponents? A negative exponent is defined as the multiplicative inverse of the base, raised to the power which is opposite to the given power. In simple words, we write the reciprocal of the number and then solve it like positive exponents. For example, (2/3)-2 can be written as (3/2)2.
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| 1. | What are Negative Exponents? |
| 2. | Negative Exponent Rules |
| 3. | Why are Negative Exponents Fractions? |
| 4. | Multiplying Negative Exponents |
| 5. | How to Solve Negative Exponents? |
| 6. | FAQs on Negative Exponents |
What are Negative Exponents?
We know that the exponent of a number tells us how many times we should multiply the base. For example, consider 82, 8 is the base, and 2 is the exponent. We know that 82 = 8 × 8. A negative exponent tells us, how many times we have to multiply the reciprocal of the base. Consider the 8-2, here, the base is 8 and we have a negative exponent (-2). 8-2 is expressed as 1/82 = 1/8×1/8.
Numbers and Expressions with Negative Exponents
Here are a few examples which express negative exponents with variables and numbers. Observe the table to see how the number is written in its reciprocal form and how the sign of the powers changes.
| 2-1 | 1/2 |
| 3-2 | 1/32=1/9 |
| x-3 | 1/x3 |
| (2 + 4x)-2 | 1/(2+4x)2 |
| (x2+ y2)-3 | 1/(x2+y2)3 |
Negative Exponent Rules
We have a set of rules or laws for negative exponents which make the process of simplification easy. Given below are the basic rules for solving negative exponents.
Rule 1: The negative exponent rule states that for every number 'a' with the negative exponent -n, take the reciprocal of the base and multiply it according to the value of the exponent: a(-n)=1/an=1/a×1/a×....n timesRule 2: The rule for a negative exponent in the denominator suggests that for every number 'a' in the denominator and its negative exponent -n, the result can be written as: 1/a(-n)=an=a×a×....n times
Let us apply these rules and see how they work with numbers.
Example 1: Solve: 2-2 + 3-2
Solution:
Use the negative exponent rule a-n=1/an2-2 + 3-2 = 1/22 + 1/32 = 1/4 + 1/9Therefore, 2-2 + 3-2 = 13/36
Example 2: Solve: 1/4-2 + 1/2-3
Solution:
Use the second rule with a negative exponent in the denominator: 1/a-n =an1/4-2 + 1/2-3 = 42 + 23 =16 + 8 = 24Therefore, 1/4-2 + 1/2-3 = 24.
Why are Negative Exponents Fractions?
A negative exponent takes us to the inverse of the number. In other words, a-n = 1/an and 5-3 becomes 1/53 = 1/125. This is how negative exponents change the numbers to fractions. Let us take another example to see how negative exponents change to fractions.
Example: Solve 2-1 + 4-2
Solution:
2-1 can be written as 1/2 and 4-2 is written as 1/42. Therefore, negative exponents get changed to fractions when the sign of their exponent changes.
Multiplying Negative Exponents
Multiplication of negative exponents is the same as the multiplication of any other number. As we have already discussed that negative exponents can be expressed as fractions, so they can easily be solved after they are converted to fractions. After this conversion, we multiply negative exponents using the same rules that we apply for multiplying positive exponents. Let's understand the multiplication of negative exponents with the following example.
Example: Solve: (4/5)-3 × (10/3)-2
The first step is to write the expression in its reciprocal form, which changes the negative exponent to a positive one: (5/4)3×(3/10)2Now open the brackets: (frac5^3 imes 3^24^3 imes 10^2)(∵102=(5×2)2 =52×22)Check the common base and simplify: (frac5^3 imes 3^2 imes 5^-24^3 imes 2^2)(frac5 imes 3^24^3 imes 4)45/44 = 45/256How to Solve Negative Exponents?
Solving any equation or expression is all about operating on those equations or expressions. Similarly, solving negative exponents is about the simplification of terms with negative exponents and then applying the given arithmetic operations.
Solution:
First, we convert all the negative exponents to positive exponents and then simplify
Given: (frac7^3 imes 3^-421^-2)Convert the negative exponents to positive by writing the reciprocal of the particular number:(frac7^3 imes 21^23^4)Use the rule: (ab)n = an × bn and split the required number (21).(frac7^3 imes 7^2 imes 3^23^4)Use the rule: am × an = a(m+n) to combine the common base (7).75/32 =16807/9Important Notes:
Note the following points which should be remembered while we work with negative exponents.
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