Simplify the Expression: b^6 × b^3 × b^5 Using Exponent Properties

Q: Why do I add the exponents instead of multiplying them?

Because exponents tell you how many times to multiply the base! $ b^6 \times b^3 $ means 6 b's times 3 b's = 9 total b's = $ b^9 $. Adding counts the total factors correctly.

Q: What's the difference between the product rule and power rule?

Product rule: Same bases multiply → add exponents: $ b^2 \times b^3 = b^5 $Power rule: Power raised to power → multiply exponents: $ (b^2)^3 = b^6 $

Q: Can I use this rule with different bases?

No! The product rule only works with identical bases. For $ a^2 \times b^3 $, you cannot combine them - they stay as separate terms.

Q: What if I have more than three terms to multiply?

Same rule applies! Just add all the exponents together. For $ b^2 \times b^4 \times b^1 \times b^7 $, you get $ b^{2+4+1+7} = b^{14} $.

Q: How can I remember this rule?

Think of it as counting factors! $ b^3 $ means three b's, $ b^5 $ means five b's. When you multiply them, you have 3+5=8 total b's, so $ b^8 $!

Product Rules with Multiple Same Bases

Reduce the following equation:

$b^6\times b^3\times b^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, the multiplication of exponents with equal bases(A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:13 We'll apply this formula to our exercise

00:17 We'll maintain the base and add the exponents together

00:28 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$b^6\times b^3\times b^5=$

Final Answer

$b^{6+3+5}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: $b^6 \times b^3 \times b^5 = b^{6+3+5} = b^{14}$
Check: Count total factors: b appears 6+3+5=14 times total ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply 6×3×5 to get $b^{90}$ = completely wrong answer! This confuses the product rule with the power rule. Always add exponents when multiplying same bases: 6+3+5=14.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

Because exponents tell you how many times to multiply the base! $b^6 \times b^3$ means 6 b's times 3 b's = 9 total b's = $b^9$ . Adding counts the total factors correctly.

What's the difference between the product rule and power rule?

Product rule: Same bases multiply → add exponents: $b^2 \times b^3 = b^5$
Power rule: Power raised to power → multiply exponents: $(b^2)^3 = b^6$

Can I use this rule with different bases?

No! The product rule only works with identical bases. For $a^2 \times b^3$ , you cannot combine them - they stay as separate terms.

What if I have more than three terms to multiply?

Same rule applies! Just add all the exponents together. For $b^2 \times b^4 \times b^1 \times b^7$ , you get $b^{2+4+1+7} = b^{14}$ .

How can I remember this rule?

Think of it as counting factors! $b^3$ means three b's, $b^5$ means five b's. When you multiply them, you have 3+5=8 total b's, so $b^8$ !

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