Negative Exponents

Negative Exponents are the exponents with negative values. In other words, negative exponents are the reciprocal of the exponent with similar positive values, i.e. a^-n (a negative exponent) can be understood as the reciprocal exponent as 1/aⁿ.

For example, find the value of (1/2)^-2 we can write this exponent as, (2/1)² this can be further simplified as, 4/1 or 4.

Let us learn more about what are negative exponents, their examples with solutions, negative exponents worksheet, and others in detail in this article.

What are Negative Exponents?

Negative exponents as the name suggests are the exponent with negative values and they can be easily simplified and then solved using the basic rule of exponent after taking their reciprocal. 

Negative Exponents Definition

Negative exponents represent the reciprocal of a number raised to a positive exponent.

Specifically, if a number is raised to a negative exponent, it means that the number should be divided by itself raised to the positive form of that exponent.

We now can define the negative exponent as the reciprocal exponent and this can be understood as,

a^-n = 1/aⁿ

Thus, negative exponents are easily solved using the above formula. We can understand this with the help of the following example,

Example: Simplify (2/3)^-3.

Solution:

Given, (2/3)^-3

= [1/(2/3)]³
= (3/2)³
= 3³/2³
= 27/8

Representation of Negative Exponents

The image added below shows the negative exponents formulas,

Negative Exponent Formula

The formulas used for solving the negative exponent are,

• a^-n = 1/aⁿ
• 1/a^-n = aⁿ

Apart from these formulas, all the basic formulas of the exponents are used in simplifying negative exponents.

Expressions with Negative Exponents

Expression with Negative Exponnet can easily be solved using its exponent values. Some of the examples of negative exponents are discussed in the table below,

Negative Exponent Rules

Negative exponents use all the basic exponent rules and other than that there are two basic negative exponents rules which are used to simplify the negative exponents which are,

Rule 1: We can simplify the negative exponent by first taking the reciprocal of the base and then solving for the positive power of the base, i.e. if the negative exponent a^-n is to be solved we first take the reciprocal of the base (1/a) and then solve it for its n^th power. This can be easily understood as,

a^(-n) = 1/a × 1/a × ... n times = 1/aⁿ

Rule 2: In this rule, we tend to solve for a negative exponent in the denominator if there is a negative exponent in the denominator then it is transported to the numerator and its negative sign is removed. We can understand this by the formula,

1/a^(-n) = a × a × ... .n times = aⁿ

The negative exponent's rule can be easily understood with the example discussed in the image below,

These rules can be easily understood by the example discussed below,

Example: Simplify 3^-3 × 1/(4^-2)

Solution:

Using the above rule for solving negative exponents,

a^-n = 1/aⁿ and 1/a^(-n) = aⁿ

3^-3 = 1/3³ 

     = 1/27

1/(4^-2) = 4²

          = 16

Now, 3^-3 × 1/(4^-2)

= 1/27 × 16

= 16/27

Negative Exponents are Fractions

As we already know, the negative exponent finds the reciprocal of the number and thus, we can express the negative exponent as the fractions.

This relation is expressed as, a^-n = 1/aⁿ. Thus, it is evident that negative exponents are easily considered as fractions. This can be understood by the example.

Example: Write in fraction form 3^-1 and 5^-2

Solution:

As we know that we can easily express the negative exponent as a fraction thus,

• 3^-1 = 1/3¹ = 1/3
• 5^-2 = 1/5² = 1/25

Negative Fraction Exponents

Now while solving for the exponents we easily encounter situations where the exponent part of the function is a negative fraction and thus it is very important to solve for negative fraction exponents.

For example, if we encounter a situation where we have to solve 125^-1/3 this can be understood as 1/125^1/3 now we can easily calculate the cube root for 125 in the denominator which gives the solution to our initial problem. The complete solution for this is,

125^-1/3
= 1/125^1/3
= 1/(5³)^1/3
= 1/5

Thus, the solution for 125^-1/3 is 1/5

Multiplying Negative Exponents

We can easily multiply the negative exponents as we multiply the normal exponent. We can also convert the negative exponent into a fraction and then find its multiple which can be easily solved.

We can understand the multiplication of negative exponents with the help of the example discussed below,

Example: Simplify (1/5)^-2 × (3)^-3

Solution:

= (1/5)^-2 × (3)^-3

We first solve the negative exponent by changing it to their reciprocals as,

= (5/1)² × (1/3)³

Now we will simplify each exponent individually

= 5² × 1/3³

= 25 × 1/27

Then, multiplying the fractions we get,

= 25/27

This is the required solution for the given exponent.

How to Solve Negative Exponents?

The formula used for solving the negative exponent is,

• a^-n = 1/aⁿ
• 1/a^-n = aⁿ

We can easily solve the negative exponents by following the steps below,

Step 1: Remove all the negative exponents by using the formula for negative exponent as discussed above.

Step 2: Using basic laws of exponents simplifies the remaining expression.

Step 3: Using exponent formulas write all the values in fraction form.

Step 4: Simplify and write the answer in the simplest form.

Example: Simplify (4^-3) × (2^-4/12^-3)

Solution:

Given expression: (4^-3) × (2^-4/12^-3)

Using the negative exponents formula and removing the negative exponent.

= 1/4³ × (12³/2⁴)

Using Basic Laws of Exponents

= (1/64)×(1728/16)

Simplifying,

= 108/64

In the simplest form

= 27/16

This is the simplified form of the given expression.

Articles related to Negative Exponents:

• Laws of Exponents
• Adding and Subtracting Exponents
• Multiplication and Division of Exponents

Negative Exponents Examples with Solutions

Example 1: Find the values of 7^-3 × 7³.

Solution:

7^-3 × 7³

Using the negative rule of the exponent,

1/7³ × 7³

= 7³/7³

= 1

The required solution is, 1

Example 2: Simplify for the value of x in 16/2^-x = 64.

Solution:

Given, 16/2^-x = 64

Using the negative rule of the exponent,

16×2^x = 64

Changing 16 and 64 to the power of 2

2⁴×2^x = 2⁶        [As, 2⁴ = 16, and 2⁶ = 64]

Using the exponent rule,

2^4+x = 2⁶

Now the exponents are the same as the bases are same so their power must also be the same.

4+x = 6

x = 6-4 = 2

Now the value of the x is,

x = 2

Example 3: Simplify (4/3)^-3 + (11/2)^-1.

Solution:

Using Negative Exponent Rules,

(4/3)^-3 = (3/4)³

(11/2)^-1 = (2/11)¹

= (4/3)^-3 + (2/11)¹

= (3/4)³ + (2/11)¹

= 27/64 + 2/11

= (27×11 + 2×64)/ 64×11

= (297 + 128)/704

= 425/704

The required solution is, 425/704

Negative Exponents Worksheet

1. Evaluate 2^-3

2. Simplify 5x^-2

3. Compute 3^-2

4. Simplify \frac{1}{4x^{-1}}

5. Evaluate 10^-4

6. Simplify \frac{1}{{2y^{-3}}}

7. Compute a^-4 × a²