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Negative Exponents
A negative exponent is defined as the multiplicative inverse of the base, raised to the power which is of the opposite sign of 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}. We know that an exponent refers to the number of times a number is multiplied by itself. For example, 3^{2} = 3 × 3. In the case of positive exponents, we easily multiply the number (base) by itself, but in case of negative exponents, we multiply the reciprocal of the number by itself. For example, 3^{2} = 1/3 × 1/3.
Let us learn more about negative exponents along with related rules and solve more examples.
What are Negative Exponents?
We know that the exponent of a number tells us how many times we should multiply the base. For example, in 8^{2}, 8 is the base, and 2 is the exponent. We know that 8^{2} = 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 1/8 × 1/8 = 1/8^{2}.
Numbers and Expressions with Negative Exponents
Here are a few examples which express negative exponents with variables and numbers. Observe the table given below to see how the number/expression with a negative exponent is written in its reciprocal form and how the sign of the powers changes.
Negative Exponent  Result 

2^{1}  1/2 
3^{2}  1/3^{2 }= 1/9 
x^{3}  1/x^{3} 
(2 + 4x)^{2}  1/(2 + 4x)^{2} 
(x^{2} + y^{2})^{3}  1/(x^{2} + y^{2})^{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 a base 'a' with the negative exponent n, take the reciprocal of the base (which is 1/a) and multiply it by itself n times.
i.e., a^{(n)} = 1/a × 1/a × ... n times = 1/a^{n}  Rule 2: The rule is the same even when there is a negative exponent in the denominator.
i.e., 1/a^{(n)} = a × a × ... .n times = a^{n}
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/a^{n}
 2^{2 }+ 3^{2} = 1/2^{2 }+ 1/3^{2} = 1/4 + 1/9
 Take the Least Common Multiple (LCM): (9 + 4)/36 = 13/36
Therefore, 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 }=a^{n}
 1/4^{2 }+ 1/2^{3 }= 4^{2 }+ 2^{3} =16 + 8 = 24
Therefore, 1/4^{2} + 1/2^{3} = 24.
Negative Exponents are Fractions
A negative exponent takes us to the inverse of the number. In other words, a^{n} = 1/a^{n} and 5^{3} becomes 1/5^{3} = 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: Express 2^{1} and 4^{2} as fractions.
Solution:
2^{1} can be written as 1/2 and 4^{2} is written as 1/4^{2}. Therefore, negative exponents get changed to fractions when the sign of their exponent changes.
Negative Fraction Exponents
Sometimes, we might have a negative fractional exponent like 4^{3/2}. We can apply the same rule a^{n} = 1/a^{n} to express this in terms of a positive exponent. i.e., 4^{3/2} = 1/4^{3/2}. Further, we can simplify this using the exponent rules.
4^{3/2} = 1/4^{3/2}
= 1 / (2^{2})^{3/2}
= 1 / 2^{3}
= 1/8
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 multiplication rule that we apply for multiplying positive exponents. Let us 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)^{2}
 Now open the brackets: \(\frac{5^{3} \times 3^{2}}{4^{3} \times 10^{2}}\)
 We know that 10^{2}=(5×2)^{2 }=5^{2}×2^{2}, so we can substitute 10^{2 }by 5^{2}×2^{2}. Then we will check the common base and simplify: \(\frac{5^{3} \times 3^{2} \times 5^{2}}{4^{3} \times 2^{2}}\)
 \(\frac{5 \times 3^{2}}{4^{3} \times 4}\)
 45/4^{4 }= 45/256
How to Solve Negative Exponents?
To solve expressions involving negative exponents, first convert them into positive exponents using one of the following rules and simplify:
 a^{n} = 1/a^{n}
 1/a^{n} = a^{n}
Example: Solve: (7^{3}) × (3^{4}/21^{2})
Solution:
First, we convert all the negative exponents to positive exponents and then simplify.
 Given: \(\frac{7^{3} \times 3^{4}}{21^{2}}\)
 Convert the negative exponents to positive by applying the above rules:\(\frac{7^{3} \times 21^{2}}{3^{4}}\)
 Use the rule: (ab)^{n} = a^{n} × b^{n} and split the required number (21).
 \(\frac{7^{3} \times 7^{2} \times 3^{2}}{3^{4}}\)
 Use the rule: a^{m }× a^{n }= a^{(m+n)} to combine the common base (7).
 7^{5}/3^{2} =16807/9
Important Notes on Negative Exponents:
 Exponent or power means the number of times the base needs to be multiplied by itself.
a^{m} = a × a × a ….. m times
a^{m} = 1/a × 1/a × 1/a ….. m times  a^{n} is also known as the multiplicative inverse of a^{n}.
 If a^{m} = a^{n} then m = n.
 The relation between the exponent (positive powers) and the negative exponent (negative power) is expressed as a^{x}=1/a^{x}
☛ Related Topics:
Examples of Negative Exponents

Example 1: Find the solution of the given expression (3^{2} + 4^{2})^{2} .
Solution:
The given expression is,
(3^{2} + 4^{2})^{2} = (9 + 16)^{2}
= (25)^{2}
= 1/25^{2} (by negative exponents rule)
= 1/625.
Therefore, (3^{2} + 4^{2})^{2} = 1/625Answer: 1/625

Example 2: Find the value of x in 27/3^{x} = 3^{6}
Solution:
Here we have negative exponents with variables.
27/3^{x} = 3^{6}
3^{3}/3^{x }= 3^{6}
3^{3 } × 3^{x}= 3^{6 }
3^{(3 + x) }= 3^{6}If bases are the same then exponents must be equal, so, 3 + x = 6. Solving this, x = 3.
Answer: x = 3.

Example 3: Simplify the following using negative exponent rules: (2/3)^{2} + (5)^{1}
Solution:
By using negative exponent rules, we can write (2/3)^{2} as (3/2)^{2} and (5)^{1} as 1/5. So, we can simplify the given expression as,
= (3/2)^{2} + 1/5
= 9/4 + 1/5
After taking the LCM, we get, (45 + 4)/20
49/20
Therefore, (2/3)^{2} + (5)^{1} is simplified to 49/20.Answer: 49/20.
FAQs on Negative Exponents
What do Negative Exponents Mean?
The negative exponents mean the negative numbers that are present in place of exponents. For example, in the number 2^{8}, 8 is the negative exponent of base 2.
Do negative exponents Result in Negative Numbers?
No, it is not necessary that negative exponents give negative numbers. For example, 2^{3} = 1/8, which is a positive number.
How to Calculate Negative Exponents?
Negative exponents are calculated using the same laws of exponents that are used to solve positive exponents. For example, to solve: 3^{3} + 1/2^{4}, first we change these to their reciprocal form: 1/3^{3} + 2^{4}, then simplify 1/27 + 16. Taking the LCM, [1+ (16 × 27)]/27 = 433/27.
What is the Rule for Negative Exponents?
There are two main rules that are helpful when dealing with negative exponents:
 a^{n} = 1/a^{n}
 1/a^{n} = a^{n}
How to Solve Fractions with Negative Exponents?
Fractions with negative exponents can be solved by taking the reciprocal of the fraction. Then, find the value of the number by taking the positive value of the given negative exponent. For example, (3/4)^{2} = (4/3)^{2} = 4^{2}/3^{2}. This results in 16/9 which is the final answer.
How to Divide Negative Exponents?
Dividing exponents with the same base results in the subtraction of exponents. For example, to solve y^{5 }÷ y^{3} = y^{5(3) }= y^{8}. This can be simplified in an alternative way also. i.e., y^{5 }÷ y^{3} = y^{5}/y^{3}, first we change the negative exponent (y^{3}) to a positive one by writing its reciprocal. This makes it: y^{5} × y^{3} = y^{(5+3)} = y^{8}.
How to Multiply Negative Exponents?
While multiplying negative exponents, first we need to convert them to positive exponents by writing the respective numbers in their reciprocal form. Once they are converted to positive ones, we multiply them using the same rules that we apply for multiplying positive exponents. For example, y^{5} × y^{2} = 1/y^{5} × 1/y^{2 }= 1/y^{(5+2) }= 1/y^{7}
Why are Negative Exponents Reciprocals?
When we need to change a negative exponent to a positive one, we are supposed to write the reciprocal of the given number. So, the negative sign on an exponent indirectly means the reciprocal of the given number, in the same way as a positive exponent means the repeated multiplication of the base.
What is 10 to the Negative Power of 2?
10 to the negative power of 2 is represented as 10^{2}, which is equal to (1/10^{2}) = 1/100.
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