Multiplication of Exponents

Multiplication of exponents, commonly known as the Product Rule, is a mathematical operation used to combine terms that share the same base. When you multiply two exponential expressions with identical bases, you retain that base and add the exponents together to determine the final power. This simplifies complex algebraic and numerical expressions into a single term.

Examples:

• \(2^3 \cdot 2^4 = 2^7\)
   
• \(x^5 \cdot x^2 = x^7\)
   
• \(10^2 \cdot 10^6 = 10^8\)
   
• \(y^4 \cdot y^{-3} = y^1\) (or simply y)
   
• \(3^2 \cdot 3^3 = 3^5\)

Exponents are a concise mathematical notation for expressing repeated multiplication, where a "base" number is multiplied by itself a specific number of times, as indicated by the superscript. This notation simplifies the representation of large values and patterns without requiring long factor strings. Clear instances of this concept include \(2^5 = 32, 10^3 = 1000, 4^3 = 64\), and \(9^2 = 81\).

In multiplying exponents with the same base, we keep the base the same and only the exponents get added. Let a^p and a^q be the two terms, where ‘a’ is the base and the exponents are ‘p’ and ‘q.’

If the terms are given as a^p and a^q, we multiply them as: 
\(a^p \times a^q = a^{(p+q)} \)

Example:

Find the product of \(\mathbf{3^2 \times 5^2}\)

Now substitute the values in the expression \(a^p \times b^p = (a \times b)^p\):

\(3^2 \times 5^2 = (3 \times 5)^2 = 15^2\)

Since \(15^2\) is '\(15 \times 15\)', we get the product as 225

Multiplying exponents with different bases depends on two scenarios:
 

Different bases with the same exponent

For terms with different bases and the same exponent, the expression will be in the form: \( a^p x b^p\). Here, ‘a’ and ‘b’ are the bases, where ‘p’ will be the exponent for both the terms. To get the product, first multiply the base and then apply the exponent. Hence, we can write \(a^p \times b^p = (a \times b)^p \). 

Example: 

Find the product of 2² × 4²

Now substitute the values in the expression \(a^p \times b^p = (a \times b)^p \):
 

\(2^2 \times 4^2 = (2 \times 4)^2 = 8^2 \)

Since 8² is ‘\(8 × 8\)’, we get the product as 64

Different bases with different exponents

When the given terms differ in both base and exponent, they are solved separately. If the given terms are a^p and b^q, their product will be (a^p) × (b^q).

Example:

Find the product of \((4^2) \times (5^3) \)

First, find the values of 4² and 5³. After finding the product of each, multiply them together to obtain the final product.

\(4^2 = 4 \times 4 = 16 \)

\(5^3 = 5 \times 5 \times 5 = 125 \)

Therefore,
 

\(4^2 \times 5^3 = 16 \times 125 \)

\(16 × 125 = 2000\)

While multiplying terms with negative exponents, We can take the reciprocal of the base. For example,\(\) 4^-2 can be expressed as \(\frac{1}{4^2} \) 

There are three cases to be followed while you multiply the terms with negative exponents. Let’s discuss them in more detail.
 

Case 1: The formula to be used when we have the same bases, but different negative exponents is \(a^{-p} \times a^{-q} = \frac{1}{a^{p+q}} \)
  
Example:

Find the product of \(2^{-1} \times 2^{-2} \)

According to the formula, we find the product as

\(2^{-1} \times 2^{-2} = 2^{-(1+2)} = 2^{-3} = \frac{1}{8} = 0.125 \)

Case 2: When the bases are different, but the negative exponents are the same,
use the formula \(a^{-p} \times b^{-p} = \frac{1}{(a \times b)^p} \)

Example:

Find the product of \(3^{-2} \) and \(4^{-2} \)

According to the formula, we find the product as 

\(3^{-2} \times 4^{-2} = \frac{1}{(3 \times 4)^2} = \frac{1}{12^2} = \frac{1}{144} \approx 0.006944 \)

Case 3: When both the bases and negative exponents are different, use the formula \(a^{-p} \times b^{-q} = \frac{1}{a^p} \times \frac{1}{b^q} \)’

Example:

Multiply \(2^{-2}\) and \(3^{-4}\)

According to the formula, we find the product as

\(2^{-2} \times 3^{-4} = \frac{1}{2^2} \times \frac{1}{3^4} = \frac{1}{4} \times \frac{1}{81} = \frac{1}{4 \times 81} = \frac{1}{324} \approx 0.00309 \)

The same exponent rules apply when we have to multiply exponents with variables.

• Just add the exponents when the bases are the same.
  
  Example:
  Find the product of a² and a³ → \(a^2 \times a^3 = a^{2+3} = a^5 \)

• We should first multiply the bases when the exponents are the same, but the bases are different.
  
  Example:
  Find the product of a⁵ and b⁵ → \(a^5 \times b^5 = (a \times b)^5 \)

• If both the variables and the exponents are different, evaluate them separately.
  
  Example:
  Find the product of a³ and b⁹ → \(a^3 \times b^9 = a^3 b^9 \)

While multiplying the exponents with a square root, the base remains a square root and the same exponential rules apply. The ‘√’ symbol is used to express the square root of a number. Hence, √b can be written in its radical form as b^1/2.

To write the given exponential expression into radical exponent, multiply the exponent with ½.  For example, (√b)n is expressed as (b^1/2)ⁿ which gives b^n/2.

Given below are the rules to be followed while multiplying the exponents with square roots.

Rule 1: Add the exponents when both the square root bases are the same

Example:
Find the product of  (√3)² and (√3)³

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

We know that \((\sqrt{b})^n = (b^{1/2})^n = b^{n/2} \). Therefore, (√3)⁵ can be expressed as (√3)^5/2.

Rule 2: If the exponents are the same and the square root bases are different, the bases are multiplied first

Example:
Find the product of  (√5)² and (√2)²

\((\sqrt{5})^2 \times (\sqrt{2})^2 = (\sqrt{5} \times \sqrt{2})^2 = (\sqrt{10})^2 = 10^{2/2} = 10^1 = 10 \)

Rule 3: If the base of the square root is different from the exponent, then they are calculated separately and then multiplied together.

Example:
Find the product of (√5)³ and (√3)⁵

\((\sqrt{5})^3 = 5^{3/2} \)
 

\((\sqrt{3})^5 = 3^{5/2} \)
 

So the product will be \((\sqrt{5})^2 \times (\sqrt{3})^5 \)

Here, the base will be a fraction. For e.g., a/bⁿ, where ‘a/b’ is the base and ‘n’ is the exponent.

The rules to multiply the exponents with fractions are given below:

• Rule 1: When the bases are the same, just add the exponents.
  
  \(\left(\frac{a}{b}\right)^m \times \left(\frac{a}{b}\right)^n = \left(\frac{a}{b}\right)^{m+n} \)
  
  Example:
  Find the product of \(\left(\frac{2}{4}\right)^6 \times \left(\frac{2}{4}\right)^4 \)
  
  \(\left(\frac{2}{4}\right)^6 \times \left(\frac{2}{4}\right)^4 = \left(\frac{2}{4}\right)^{6+4} = \left(\frac{2}{4}\right)^{10} = \left(\frac{1}{2}\right)^{10} \)

• Rule 2: When the fractional bases are different and the exponents are the same, the bases get evaluated first
  
  \(\left(\frac{a}{b}\right)^n \times \left(\frac{c}{d}\right)^n = \left[\left(\frac{a}{b}\right) \times \left(\frac{c}{d}\right)\right]^n \)
  
  Example:
  Find the product of \(\left(\frac{1}{2}\right)^4 \times \left(\frac{6}{12}\right)^4 \)
  
  \(\left(\frac{1}{2}\right)^4 \times \left(\frac{6}{12}\right)^4 = \left[\left(\frac{1}{2}\right) \times \left(\frac{6}{12}\right)\right]^4 = \left(\frac{1}{4}\right)^4 \)

• Rule 3: When both the fractional base and exponent are different, they are evaluated separately, that is,
  
  \(\left(\frac{a}{b}\right)^m \times \left(\frac{c}{d}\right)^n = \frac{a^m}{b^m} \times \frac{c^n}{d^n} \)
  
  Example:
  Find the product of \(\left(\frac{2}{3}\right)^3 \times \left(\frac{3}{4}\right)^2 \)
  
  \(\left(\frac{2}{3}\right)^3 \times \left(\frac{3}{4}\right)^2 = \frac{2^3}{3^3} \times \frac{3^2}{4^2} = \frac{8}{27} \times \frac{9}{16} = \frac{72}{432} = \frac{1}{6} \)

Fractional exponents refer to a term’s exponent in a fractional form. Let’s consider the term ‘am/n’. Here, ‘m/n’ is the exponent.

Check the given table to understand the rules applied while multiplying.
 

The concept of exponents can often feel abstract because it compresses large mathematical operations into tiny superscripts. When students first encounter these problems, the instinct is usually to multiply the numbers they see rather than apply the specific rules of the operation. To bridge the gap between rote memorization and proper understanding, here are a few tips and tricks to help reinforce the logic behind the math.

• The "Expand It Out" Strategy: When in doubt, write it out. If a student forgets the rule for multiplication of exponents, have them write the full expansion. Seeing that \(2^3 \cdot 2^2\) is actually \((2 \cdot 2 \cdot 2) \cdot (2 \cdot 2)\) makes counting the five 2s intuitive and proves why addition is the correct step.
   
• Color-Code the Components: Visual distinction helps to separate the base from the exponent. Use one color (like purple) for the base and a contrasting color (like orange) for the exponent. This makes it easier to spot when bases match—a crucial step in learning how to multiply exponents correctly.
   
• Focus on the "Same Base" Rule: Before doing any math, establish a "Base Check" habit. Remind students that how to multiply powers effectively depends entirely on identifying if the big numbers (bases) are identical. If they don't match, the simple addition rule doesn't apply.
   
• Use Small Numbers First: Avoid complicating the process with large arithmetic. Start with bases like two or x. Using small, manageable numbers keeps the focus strictly on the mechanics of the exponent rules, rather than getting bogged down in calculation errors.
   
• Contrast with Addition: A standard error is confusing \(x^2 \cdot x^3\) with \(x^2 + x^3\). Create side-by-side comparisons showing that multiplying exponents results in a higher power (\(x^5\)), while adding exponents results in two separate terms that cannot be combined.
   
• Create a "Power Tower" Visual: Use building blocks or Lego bricks to represent exponents physically. A stack of 3 blocks next to a stack of 4 blocks can be physically pushed together to make a stack of 7. This provides a tangible representation of how to multiply exponents by simply "gathering" the total count.
   
• The "Keep the Base" Mantra: Students often accidentally multiply the bases (turning \(2^3 \cdot 2^4\) into \(4^7\)). Use a rhythmic mantra like "Keep the base, add the space (exponents)" to reinforce that the base number serves as a static anchor during multiplication.

We use exponents in our daily life. Given below are some real-life applications. Let’s discuss them further: 

• Finance: To calculate the growth of an investment.
   
• Science: To calculate the population growth, speed of radioactive decay, etc.
   
• Others: To calculate the area and volume, to measure and use in architecture.
   
• Technology: To represent large numbers in computing, such as memory storage (bytes, kilobytes, gigabytes) using powers of 2.
   
• Medicine: To model the spread of diseases or dosage calculations over time using exponential growth or decay.