Negative Exponents
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) with 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, 3^{2} can be written as 1/3^{2}. Let us learn more about negative exponents in this lesson.
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? 
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 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^{2 }= 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 change.
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 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/a^{n}=1/a×1/a×....n times
 Rule 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)}=a^{n}=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/a^{n}
 2^{2 }+ 3^{2} = 1/2^{2 }+ 1/3^{2} = 1/4 + 1/9
 Take the Least Common Multiple (LCM): (4+9)/36 = 13/36
Example 2: Solve: 1/4^{2} + 1/2^{3}
Solution:
 Use the second rule with a negative exponent in denominator: 1/a^{n }=a^{n}
 1/4^{2 }+ 1/2^{3 }= 4^{2 }+ 2^{3} =16 + 8 = 24
Why are Negative Exponents 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: Solve 2^{1} + 4^{2}
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.
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. 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)^{2 }
 Now open the brackets: \(\frac{5^{3} \times 3^{2}}{4^{3} \times 10^{2}}\)^{ }(∵10^{2}=(5×2)^{2 }=5^{2}×2^{2})
 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?
Solving any equation or expression is all about operating on those equations or expressions. Similarly, solving exponents (or negative exponents) is about the simplification of terms with exponents (or negative exponents) and then applying the given arithmetic operations.
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 writing the reciprocal of the particular number:\(\frac{7^{3} \times 21^{2}}{3^{4}}\)
 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).
 \(\frac{7^{5} \times 3^{2}}{1}\)
 7^{5}/3^{2 }=16807/9
Important Points
Note the following points which should be remembered while we work with negative exponents.
 Exponent or power means the number of times the base needs to be multiplied with 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}
Negative Exponents Related Articles:
 Exponent Rules
 Exponents
 Multiplying Exponents
 Fractional Exponents
 Irrational Exponents
 Exponents Formula
 Exponent Equations
Solved Examples on Negative Exponents

Example 1: Find the solution for the given problem: (3^{2} + 4^{2})^{2}
Solution:
(3^{2} + 4^{2})^{2} = (9 + 16)^{2} = (25)^{2} = 1/25^{2} = 1/625. Therefore, (3^{2} + 4^{2})^{2} = 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 same then exponents must be equal x + 3 = 6, x = 3. Therefore, the value of x = 3.
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Practice Questions on Negative Exponents
FAQs on Negative Exponents
What are Negative Numbers with Exponents?
Negative numbers with exponents follow the same laws as positive numbers with exponents, the only difference is that the negative numbers give a negative result when their exponent is odd and they give a positive result when the exponent is even. For example, (5)^{3} = 125, (5)^{4} = 625.
How to Simplify Negative Exponents?
Negative exponents are simplified 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.
How to Divide Negative Exponents?
Dividing exponents with the same base is the same as multiplying exponents, but first, we need to convert them to positive exponents. We know that when the exponents with the same base are multiplied, the powers are added and we use the same rule while dividing exponents. For example, to solve y^{5 }÷ y^{3}, or, 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.