Negative Exponents

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.

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 1/8² = 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 given below to see how the number is written in its reciprocal form and how the sign of the powers changes.

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 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)=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/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.

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.

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 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

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.

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 writing the reciprocal of the particular number:\(\frac{7^{3} \times 21^{2}}{3^{4}}\)
• 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: a^m × aⁿ = a^(m+n) to combine the common base (7).
• 7⁵/3² =16807/9

Important Notes:

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 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 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 a^x=1/a^-x

☛ Related Topics

• Exponent Rules
• Multiplying Exponents
• Fractional Exponents
• Irrational Exponents
• Exponents Formula
• Exponential Equations

What are Negative Exponents?

When we have negative numbers as exponents, we call them negative exponents. For example, in the number 2^-8, -8 is the negative exponent of base 2.

Do negative exponents make negative numbers?

No, it is not necessary that negative exponents give negative numbers. Being positive or negative depends on the base of the number. 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. A positive number with a negative exponent will always give a positive number. For example, 2^-3 = 1/8, which is a positive number.

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: 1/3³ + 2⁴, 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 negative exponent rules that are given below:

• Let a be the base and n be the exponent, we have, a^-n = 1/aⁿ
• 1/a^-n = aⁿ

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 y⁵ ÷ y^-3, or, y⁵/y^-3, first we change the negative exponent (y^-3) to a positive one by writing its reciprocal. This makes it: y⁵ × y³ = y⁽⁵⁺³⁾ = y⁸.

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/y⁵ × 1/y² = 1/y⁽⁵⁺²⁾ = 1/y⁷

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.

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 can be solved by taking the reciprocal of the fraction, which is 4/3. Now, find the positive exponent value of 4/3, which is (4/3)² = 4²/3². This results in 16/9 which is the final answer.

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 (1/10²) = 1/100.