Negative Exponents - Rules, Fractions, Solve, Calculate - Cuemath

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, 32 = 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.

1.	What are Negative Exponents?
2.	Numbers and Expressions with Negative Exponents
3.	Negative Exponent Rules
4.	Why are Negative Exponents Fractions?
5.	Negative Fraction Exponents
6.	Multiplying Negative Exponents
7.	How to Solve Negative Exponents?
8.	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, in 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 1/8 × 1/8 = 1/82.

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	1/32 = 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 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/an
• 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 = an

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/an
• 2-2 + 3-2 = 1/22 + 1/32 = 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 =an
• 1/4-2 + 1/2-3 = 42 + 23 =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/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: Express 2-1 and 4-2 as fractions.

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.

Negative Fraction Exponents

Sometimes, we might have a negative fractional exponent like 4-3/2. We can apply the same rule a-n = 1/an to express this in terms of a positive exponent. i.e., 4-3/2 = 1/43/2. Further, we can simplify this using the exponent rules.

4-3/2 = 1/43/2

= 1 / (22)3/2

= 1 / 23

= 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 102=(5×2)2 =52×22, so we can substitute 102 by 52×22. Then we will check the common base and simplify: \(\frac{5^{3} \times 3^{2} \times 5^{-2}}{4^{3} \times 2^{2}}\)
• \(\frac{5 \times 3^{2}}{4^{3} \times 4}\)
• 45/44 = 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/an
• 1/a-n = an

Example: Solve: (73) × (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 = an × bn and split the required number (21).
• \(\frac{7^{3} \times 7^{2} \times 3^{2}}{3^{4}}\)
• Use the rule: am × an = a(m+n) to combine the common base (7).
• 75/32 =16807/9

Important Notes on Negative Exponents:

• Exponent or power means the number of times the base needs to be multiplied by itself. am = 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 an.
• If a-m = a-n then m = n.
• The relation between the exponent (positive powers) and the negative exponent (negative power) is expressed as ax=1/a-x

☛ Related Topics:

• Negative Exponents Calculator
• Exponent Rules Calculator
• Exponent Calculator