What is the Product Rule of Exponents for Multiple Bases?

S3-SA1-0666

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Product Rule of Exponents for Multiple Bases tells us how to multiply numbers that have different bases but the same exponent. When you multiply such numbers, you can first multiply their bases together and then raise the result to the common exponent.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, each buying 2 packets of biscuits, and each packet has 3 biscuits. Rohan buys (2^3) biscuits and Priya also buys (2^3) biscuits. If instead, Rohan buys (2^3) biscuits and Priya buys (4^3) biscuits, to find the total biscuits, we use this rule. Since both 2 and 4 are raised to the power of 3, we can multiply 2 and 4 first, then raise the answer to the power of 3.

Worked Example

Step-by-Step

Let's calculate 2^3 * 3^3.
---Step 1: Identify the bases and the common exponent. Here, the bases are 2 and 3, and the common exponent is 3.
---Step 2: According to the rule, multiply the bases first. So, 2 * 3 = 6.
---Step 3: Now, raise the product of the bases to the common exponent. So, 6^3.
---Step 4: Calculate 6^3, which means 6 * 6 * 6.
---Step 5: 6 * 6 = 36.
---Step 6: 36 * 6 = 216.
Answer: 2^3 * 3^3 = 216

Why It Matters

Understanding this rule helps simplify complex calculations in fields like data science and engineering, where you often work with very large or very small numbers. It's a fundamental concept that can lead to exciting careers in areas like creating smart apps or designing new technologies.

Common Mistakes

MISTAKE: Adding the bases instead of multiplying them. For example, calculating (2+3)^3 instead of (2*3)^3 for 2^3 * 3^3. | CORRECTION: Always multiply the bases when they are different but have the same exponent.

MISTAKE: Multiplying the exponents when the bases are different. For example, thinking 2^3 * 3^3 = (2*3)^(3*3). | CORRECTION: The exponent remains the same; only the bases are multiplied.

MISTAKE: Applying this rule when exponents are different. For example, trying to use this rule for 2^3 * 3^2. | CORRECTION: This rule only applies when the exponents are exactly the same.

Practice Questions

Try It Yourself

QUESTION: Simplify 5^2 * 4^2. | ANSWER: (5*4)^2 = 20^2 = 400

QUESTION: Calculate 10^3 * 2^3. | ANSWER: (10*2)^3 = 20^3 = 8000

QUESTION: A farmer has (2^4) cows and another farmer has (3^4) cows. If they combine their cows, and each cow costs (1^4) rupees, what is the total value if we consider the product of their cows? (Hint: The question is asking for 2^4 * 3^4). | ANSWER: (2*3)^4 = 6^4 = 1296

MCQ

Quick Quiz

Which of the following expressions is equivalent to 7^2 * 3^2?

(7+3)^2

(7*3)^2

(7*3)^(2*2)

7*3*2

The Correct Answer Is:

The product rule for exponents with multiple bases states that if the exponents are the same, you multiply the bases and keep the exponent. So, 7^2 * 3^2 = (7*3)^2.

Real World Connection

In the Real World

In computer science, when designing algorithms, you might need to calculate how quickly data grows. For instance, if a server can process 2^5 tasks per second and another server can process 3^5 tasks per second, knowing this rule helps quickly estimate their combined processing power as (2*3)^5, which is 6^5. This is used in companies like Google or Amazon to manage their vast data centers.

Key Vocabulary

Key Terms

BASE: The number that is multiplied by itself | EXPONENT: The small number written above and to the right of the base, indicating how many times the base is multiplied by itself | PRODUCT: The result of multiplication | SIMPLIFY: To make an expression or equation easier to understand or solve

What's Next

What to Learn Next

Next, you should explore the 'Quotient Rule of Exponents for Multiple Bases'. It's similar to this rule but deals with division instead of multiplication, and it will further strengthen your understanding of how exponents work.