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What is the Rule for Multiplying Exponents with Same Base?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
When you multiply numbers that have the same base and different powers (exponents), you can simply add their powers together. The base remains the same. This rule helps simplify calculations with large numbers.
Simple Example
Quick Example
Imagine you have 2^3 ladoos and your friend has 2^4 ladoos. If you combine them, how many 'base 2' ladoos do you have? Using the rule, you add the powers: 3 + 4 = 7. So, you have 2^7 ladoos in total. This is much easier than calculating 8 + 16 = 24 and then trying to figure out what power of 2 gives 24!
Worked Example
Step-by-Step
Let's multiply 5^2 by 5^3.
Step 1: Identify the base. Here, the base is 5 for both numbers.
---Step 2: Identify the powers (exponents). The powers are 2 and 3.
---Step 3: Since the bases are the same, apply the rule: add the powers. So, 2 + 3 = 5.
---Step 4: Keep the base the same and use the new sum as the power. This gives 5^5.
---Step 5: (Optional: Calculate the final value) 5^5 = 5 x 5 x 5 x 5 x 5 = 3125.
Answer: 5^2 x 5^3 = 5^5
Why It Matters
Understanding this rule is super important for anyone working with data or technology. Engineers use it to design circuits, computer scientists use it in coding, and even data scientists use it to handle massive datasets. It's a fundamental concept that helps build complex systems in areas like AI and cryptography.
Common Mistakes
MISTAKE: Multiplying the bases as well (e.g., 2^3 x 2^4 = (2x2)^(3+4) = 4^7) | CORRECTION: Only the exponents are added; the base remains the same (2^3 x 2^4 = 2^(3+4) = 2^7)
MISTAKE: Multiplying the exponents instead of adding them (e.g., 3^2 x 3^4 = 3^(2x4) = 3^8) | CORRECTION: The exponents must be added, not multiplied (3^2 x 3^4 = 3^(2+4) = 3^6)
MISTAKE: Applying the rule when bases are different (e.g., 2^3 x 3^2 = (2x3)^(3+2) = 6^5) | CORRECTION: This rule only works when the bases are exactly the same. If bases are different, you cannot combine them this way.
Practice Questions
Try It Yourself
QUESTION: Simplify: 7^5 x 7^2 | ANSWER: 7^7
QUESTION: What is the value of 10^3 x 10^1? | ANSWER: 10^4 or 10,000
QUESTION: If a^4 x a^x = a^9, what is the value of x? | ANSWER: x = 5
MCQ
Quick Quiz
Which of the following correctly simplifies 6^3 x 6^5?
6^15
36^8
6^8
12^8
The Correct Answer Is:
C
The rule for multiplying exponents with the same base is to add the powers and keep the base the same. So, 6^(3+5) = 6^8. Options A, B, and D incorrectly multiply the powers or change the base.
Real World Connection
In the Real World
Imagine a company like Flipkart or Amazon tracking the number of products sold. If they sold 10^6 items last year and 10^7 items this year, knowing this rule helps them quickly combine these figures to see total sales over two years as 10^13, without having to write out huge numbers. This is also how computer memory (like gigabytes) is calculated, using powers of 2.
Key Vocabulary
Key Terms
BASE: The number that is multiplied by itself in an exponent (e.g., in 5^3, 5 is the base) | EXPONENT/POWER: The small number written above and to the right of the base, indicating how many times the base is multiplied by itself (e.g., in 5^3, 3 is the exponent) | PRODUCT: The result of multiplying two or more numbers | SIMPLIFY: To make an expression or equation easier to understand or solve
What's Next
What to Learn Next
Great job learning this rule! Next, you should explore 'What is the Rule for Dividing Exponents with Same Base?'. It builds directly on this concept and will help you understand how to handle division with exponents, which is equally useful!
