Solve a³ × a⁴: Product Rule of Exponents Step-by-Step

Q: Why do we add the exponents when multiplying?

Think of it as counting factors! $ a^3 $ means a×a×a and $ a^4 $ means a×a×a×a. When you multiply them together, you get 3+4=7 total factors of a.

Q: What if the bases are different, like $ a^3 \times b^4 $ ?

You cannot combine them! The product rule only works with identical bases. Different bases like $ a^3 \times b^4 $ stay as they are.

Q: Does this rule work with negative exponents?

Yes! The rule $ a^m \times a^n = a^{m+n} $ works with any exponents - positive, negative, or zero. Just add them normally!

Q: How is this different from $ (a^3)^4 $ ?

That's power of a power, not multiplication! For $ (a^3)^4 $, you multiply exponents: 3×4=12, giving $ a^{12} $. Different rule entirely!

Q: Can I use this rule with more than two terms?

Absolutely! $ a^2 \times a^3 \times a^5 = a^{2+3+5} = a^{10} $. Just keep adding all the exponents when the bases are identical.

Exponent Rules with Identical Bases

$a^3\times a^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When multiplying powers with equal bases

00:06 The power of the result equals the sum of the powers

00:09 We'll apply this formula to our exercise, and add the powers together

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$a^3\times a^4=$

Step-by-step solution

Here, we will need to calculate a multiplication between terms with identical bases, therefore we will use the appropriate power property:

$b^m\cdot b^n=b^{m+n}$ Note that this property can only be used to calculate the multiplication between terms with identical bases,

We apply it to the problem:

$a^3\cdot a^4=a^{3+4}=a^7$ Therefore, the correct answer is option b.

Final Answer

$a^7$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $a^3 \times a^4 = a^{3+4} = a^7$
Check: Count total factors: a×a×a×a×a×a×a = 7 factors ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 3 × 4 = 12 to get $a^{12}$ ! This confuses multiplication of exponents with addition. The rule states you multiply the bases (which are the same) by adding exponents. Always remember: same base multiplication means add exponents.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add the exponents when multiplying?

Think of it as counting factors! $a^3$ means a×a×a and $a^4$ means a×a×a×a. When you multiply them together, you get 3+4=7 total factors of a.

What if the bases are different, like $a^3 \times b^4$ ?

You cannot combine them! The product rule only works with identical bases. Different bases like $a^3 \times b^4$ stay as they are.

Does this rule work with negative exponents?

Yes! The rule $a^m \times a^n = a^{m+n}$ works with any exponents - positive, negative, or zero. Just add them normally!

How is this different from $(a^3)^4$ ?

That's power of a power, not multiplication! For $(a^3)^4$ , you multiply exponents: 3×4=12, giving $a^{12}$ . Different rule entirely!

Can I use this rule with more than two terms?

Absolutely! $a^2 \times a^3 \times a^5 = a^{2+3+5} = a^{10}$ . Just keep adding all the exponents when the bases are identical.

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Solve a³ × a⁴: Product Rule of Exponents Step-by-Step

Exponent Rules with Identical Bases

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we add the exponents when multiplying?

What if the bases are different, like $a^3 \times b^4$ ?

Does this rule work with negative exponents?

How is this different from $(a^3)^4$ ?

Can I use this rule with more than two terms?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Simplify y^(-2) × y^7: Laws of Exponents Practice

Simplify the Expression: a¹⁰ × b⁵ × a⁻² × b³

Comparing Mathematical Expressions: Finding Greater Value When b > 1

Expression Comparison: Determining Greater Value When c>1

Solve a³ × a⁴: Product Rule of Exponents Step-by-Step

Exponent Rules with Identical Bases

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we add the exponents when multiplying?

What if the bases are different, like a3×b4 a^3 \times b^4 a3×b4?

Does this rule work with negative exponents?

How is this different from (a3)4 (a^3)^4 (a3)4?

Can I use this rule with more than two terms?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Simplify y^(-2) × y^7: Laws of Exponents Practice

Simplify the Expression: a¹⁰ × b⁵ × a⁻² × b³

Comparing Mathematical Expressions: Finding Greater Value When b > 1

Expression Comparison: Determining Greater Value When c>1

What if the bases are different, like $a^3 \times b^4$ ?

How is this different from $(a^3)^4$ ?