Multiplying and Dividing Exponents

An exponent tells us the number of times a number should be multiplied by itself. A number 'b' raised to the power 'p' can be written as: \(bᵖ = b × b × b ×... × b \)(p times) 

Where: 

• b represents any number (the base).
• p is an integer (the exponent or power).

bp is also called the pᵗʰ power of b.

As we can see, ‘b’ is multiplied by itself p times. This process is called exponentiation and is an efficient way to represent repeated multiplication.

Here are the fundamental rules of exponents:

• Product Rule: \( a^n \times a^m = a^{n+m} \)

• Quotient Rule: \( \frac{a^n}{a^m} = a^{n-m} \)

• Power Rule: \( (a^n)^m = a^{n \cdot m} \)

• Power of a Root Rule: \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \)

• Negative Exponent Rule: \( a^{-m} = \frac{1}{a^m} \)

• Zero Rule: \( a_0 = 1 \)

• One Rule: \(a_1 = a \)
   

To multiply exponents, we need to follow different rules based on whether the bases are the same or different. 

Multiplying Exponents with the Same Base

The rule for multiplying expressions with the same base is:
\( a^m \times a^n = a^{m+n} \)
Where:

‘a’ represents the common base, and ‘m’ and ‘n’ represent the exponents.

For example:  

\( 4^2 \times 4^3 = 4^{2+3} = 4^5 \)

Multiplying Exponents with Different Bases and the Same Power

When multiplying expressions with different bases but the same exponent, we will use the rule:

\( a^m \times b^m = (a \times b)^m \)

This rule works well because exponents represent repeated multiplication.

For example:

\( 5^2 \times 4^2 = (5 \times 4)^2 = 20^2 = 400 \)

Understanding the properties of exponents will help us divide exponents effectively. Let’s look at how to divide exponents using different rules:

Dividing Exponents with the Same Base

When dividing two exponential terms with the same base, we apply the quotient rule by subtracting the exponents.

Rule: \( \frac{a^m}{a^n} = a^{m-n} \)

Example: \( 8^6 \div 8^2 = 8^{6-2} = 8^4 \)

Dividing Exponents with Different Bases but the Same Exponents

When dividing the exponential terms with different bases but the same exponents:
We divide the bases and keep the exponent as it is.

Rule:

\( \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \)

For example: \( 16^2 \div 4^2 = (16 \div 4)^2 = 4^2 = 16 \)

Dividing Exponents with Coefficients

In the case of exponents with variables, we need to divide them separately and apply the exponent rules to the variable part.
For example: \( \frac{18x^2}{6x^2} \)

Let’s look at the steps:

Step 1: Divide the coefficients:

\( \frac{18}{6} = 3 \)

Step 2: Apply the quotient rule for exponents:

 \( \frac{x^2}{x^2} = x^{2-2} = x^0 = 1 \)

\(⇒ 3 ×  1 = 3 \)

The same rules that apply to numbers also apply to variables when multiplying or dividing the exponents. Let’s now go through the key rules and see how to apply them using different examples:

\( a^m \times a^n = a^{m+n} \)

\(a^m \times b^m = (a \times b)^m \)

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

\(\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m\)

Variable as the Base

We will now look at how to apply the rules for a variable as the base.

For example: Simplify \( y^2 \times (2y)^3 \)

Apply the rule and expand the expression:

\( (2y)^3 = 2^3 \times y^3 = 8y^3 \)

Multiply with \(y^2\):

\( y^2 \times 8y^3 = 8y^{2+3} = 8y^5 \)

Variable as the Exponent

When the bases are the same, we subtract the exponents using the quotient rule, even if the exponent contains a variable.

For example: Simplify \( \frac{73x + 2}{7x - 1} \)

To divide exponential terms with the same base, we subtract the exponents:

 \( \frac{7^{3x+2}}{7^{x-1}} = 7^{(3x+2) - (x-1)} = 7^{2x+3} \)

Simplify the exponent:

\( (3x + 2) - (x - 1) = 3x + 2 - x + 1 = 2x + 3 \)

\(⇒ 72x + 3 \)

Multiplication and division of exponents can often be a little confusing for students in the beginning. These tips and tricks will help guide them through these operations and ensure efficiency and accuracy.
 

• Remember \( a^m \times a^n = a^{m+n} \) and \( \frac{a^m}{a^n} = a^{m-n} \) whenever the bases are same.
   
• Handle coefficients separately. Multiply or divide the numbers in front of the variables first, then apply exponent rules to the variable parts.
   
• Always use parentheses to avoid mistakes.
   
• Break down complex expressions step-by-step to prevent errors.
   
• Double-check your final answer by substituting simple values (like 1 or 2) for the variable to ensure your result makes sense.

Multiplying and dividing exponents help us solve problems that involve large numbers or too small numbers more easily. They are widely used in different fields. Let’s now learn how they can be applied in real-life situations.

• The size of microscopic objects like cells, atoms, or distances like the speed of light can be expressed using exponents. 

           For example, \(3 × 108 \) m/s for the speed of light.

 

• Exponential formulas are used to calculate compound interest in banking.

          For example: 
          Investing $10,000 at 5% interest for 3 years:
         \( 10,000  × (1.05)3 = $11,576 \)

• Exponents help us understand everyday phenomena such as power consumption measured in watts.
  For example: For 2 hours, a 1000-watt microwave uses \(2 × 103 \) watt hours, or 2 kWh.
   
• Exponents are used in computing to represent very large or very small values, such as data storage or processing speeds.
  For example: A 1 terabyte(TB) drive = \(10^{12}\)
   
• They are also used in science to express population growth or radioactive decay through exponential equations.
  For example: \(N = N_0 \times e^{-\lambda t} \)