Negative Exponents

When a base has a negative exponent, it means we have to take the reciprocal of the number with a positive exponent.  
This can be expressed as: a - n = 1an (where a ≠ 0)
 

For example, simplifying \(8^{-2}\) gives \(\frac{1}{8^{2}}\),
simplifying \(6^{-1}\) gives \(\frac{1}{6}\), and 
simplifying \(\left(\frac{3}{4}\right)^{-2} \) gives \(\left(\frac{4}{3}\right)^2 \)
 

Image below is the representation of exponents
 

The fundamental formulas for working with negative exponents are:

\(a^{-n} = \frac{1}{a^n} \)

\(\frac{1}{a^{-n}} = a^n \)

When simplifying expressions with negative exponents, the general rules of exponents remain applicable. Let’s see how expressions with negative exponents can be solved using these principles

• \(x^{-1} = \frac{1}{x} \)
• \(x^{-2} = \frac{1}{x^2} \)
• \(x^{-n} = \frac{1}{x^n} \)
• \((x + y)^{-n} = \frac{1}{(x + y)^n} \)
• \(a^{-n} = \frac{1}{a^n} \) (a is constant here)
   

There are two primary rules for simplifying negative exponents, which build upon the basic exponent rules:

Rule 1 - Reciprocal and Positive Power: When a base has a negative power (a^-n), the reciprocal of the base   (1/a) must be taken to change the sign of the exponent from negative to positive. So, a-n becomes 1an. 

Rule 2 - Moving from Denominator to Numerator: If a base with a negative exponent is in the denominator of a fraction, then it can be moved to the numerator by changing the sign of the exponent from negative to positive. For example, (1/a^{- n}) can be written as aⁿ. This is equivalent to multiplying a by itself n times, resulting in aⁿ

The application of these negative exponent rules is further clarified by the examples provided in the image below.
 

Negative exponents means the reciprocal of a base raised to a non-negative power. 

Expressions that have negative exponents can be rewritten as fractions. 

The relationship a-n=1/an demonstrates this connection, showing that a negative exponent directly corresponds to a fractional form. 

This concept is further illustrated by the example below.
For example: Rewrite 4-2 and 2-3

Solution: We know that \(a^{-n} = \frac{1}{a^n} \),

Then, 
 \(4^{-1} = \frac{1}{4^1} = \frac{1}{4} \)
\(2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)
 

In this section, we look at how to solve expressions that have negative fractions as their exponents. Let's look at this with an example: \(8^{ \frac{-1}{3} }\) can be rewritten as \(\left(\frac{1}{8}\right)^{1/3} \) according to the property of negative exponents. Here, the exponent \(\frac{1}{3}\) indicates cube root because according to the fractional exponent rule \(a^{1/n} = \sqrt[n]{a} \). Thus, our problem reduces to finding the cube root of 8 in the denominator.

The detailed solution is:
\(8^{-1/3} \)
\(8^{-1/3} = \frac{1}{8^{1/3}} = \frac{1}{2} \)
 

The standard formulas for solving negative exponents are    :
\(a^{-n} = \frac{1}{a^n} \)
\(\frac{1}{a^{-n}} = a^n \)

Given below are the steps to solve expressions with negative exponents:

Step 1 - Eliminate Negative Signs: Apply the rules a-n=1/an and 1/a-n=an to convert all
negative exponents to positive ones.

Step 2 - Simplify Using Exponent Laws: Use the fundamental laws of exponents (like the product rule, quotient rule, power rule, etc.) to simplify the resulting expression.

Step 3 - Express in Fractional Form (if necessary):
Rewrite the expression so that all terms are in fraction form.

Step 4 - Final Simplification: Simplify the fraction to its lowest terms to obtain the final answer.

Example: Simplify \((3^{-2}) \times (4^{-3}) \)

Solution: Given \((3^{-2}) \times (4^{-3}) \)

Eliminating negative exponents: \(\frac{1}{3^2} \times \frac{1}{4^3} \)

simplifying using exponent laws:\(\frac{1}{9} \times \frac{1}{64} \)

final simplification: \(\frac{1}{576} \)

Now, let’s look at how bases with negative exponents are multiplied. To make it easy to understand, let’s approach this with an example.

Solve: \((4^{-2}) \times (1/5)^{-3} \)

Solution: Given, \((4^{-2}) \times (5^{-3}) \)
First, we address the negative exponents by converting the terms to their reciprocals:
\(= \frac{1}{4^2} \times \frac{5^3}{1} \)

Next, we simplify each term with its respective exponent:
\(= \frac{1}{4^2} \cdot 5^3 = \frac{1}{16} \cdot 125 \)

Finally, we multiply these results together:
\(= \frac{125}{16} \)
 

Let's go through the step-by-step explanation of dividing negative exponents with an example

Example: \(\frac{a-7}{ a-4}\)
Solution: Given   \(\frac{a-7}{ a-4}\)
First, we should apply the quotient rule, which is \(\frac{a^m}{a^n} = a^{m-n} \)
Here, m = -7 and n = -4
\(\frac{a^{-7}}{a^{-4}} = a^{-7-(-4)} \)
Second, simplify the subtraction
\(a^{-7-(-4)} = a^{-7+4}\)
Finally, we get the simplified answer
\(a^{-7+4} = a^{-3}\)

Students may find negative exponents confusing at first glance, but these tips and tricks can help them understand and apply the concept better.
 

• Always remember \(a^{-n} = \frac{1}{a^n}\). Negative exponents flip the base.
• Before simplifying, convert negative exponents to positive.

• Apply exponent rules; multiply by adding and divide by subtracting powers.

• Keep fractions simple by moving terms with negative exponents across the numerator and denominator.

• Practice rewriting complex expressions for better understanding.

Negative exponents can do much more than just help us score good grades. In fact, they have many real-life applications, some of which are mentioned below:
 

• Negative exponents express minuscule values like 1 × 10^-6 meter size of a bacterium. Since microorganisms are tiny, we need negative exponents to measure them for scientific purposes.

• Fractions of a second, such as milliseconds, are represented with negative exponents. For example, 1 millisecond = 10^-3 seconds.

• Small concentrations like 1 x 10^-7 mol/L hydrogen ion concentration in water are expressed using negative exponents in chemistry.

• The rapid speeds of computer processors, with operation times around 10^-9 seconds, are expressed using negative exponents.

• The short durations between wave cycles are represented with negative exponents. E.g., the time taken for a sound wave cycle is 10^-6 seconds.