Simplify the Expression: a⁴ × a⁵ × a⁶ Using Exponent Rules

Exponent Multiplication with Same Base

Reduce the following equation:

a4×a5×a6= a^4\times a^5\times a^6=

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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, multiplying exponents with the same base (A)
00:08 equals the same base raised to the sum of the exponents (N+M)
00:12 We will apply this formula to our exercise
00:16 We'll maintain the base and add the exponents together
00:42 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Reduce the following equation:

a4×a5×a6= a^4\times a^5\times a^6=

3

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic
  • Product Rule: When multiplying same bases, add the exponents
  • Technique: a4×a5×a6=a4+5+6=a15 a^4 \times a^5 \times a^6 = a^{4+5+6} = a^{15}
  • Check: Count total factors: a appears 4+5+6=15 times total ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying the exponents instead of adding them
    Don't calculate a4×5×6=a120 a^{4\times5\times6} = a^{120} ! This gives a massive wrong answer because you're raising to the power of a product instead of adding powers. Always add exponents when multiplying same bases: a4×a5×a6=a4+5+6=a15 a^4 \times a^5 \times a^6 = a^{4+5+6} = a^{15} .

Practice Quiz

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\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why do we add the exponents instead of multiplying them?

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Because exponents represent repeated multiplication! a4×a5 a^4 \times a^5 means (a×a×a×a) × (a×a×a×a×a) = 9 total a's = a9 a^9 . Adding exponents counts the total factors.

What if the bases were different, like a4×b5 a^4 \times b^5 ?

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You cannot combine them! The product rule only works with the same base. a4×b5 a^4 \times b^5 stays as a4b5 a^4b^5 - no simplification possible.

Is a4+5+6 a^{4+5+6} considered fully simplified?

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No, you should always calculate the final exponent! a4+5+6=a15 a^{4+5+6} = a^{15} is the complete simplified form. Leaving it as a4+5+6 a^{4+5+6} shows your work but isn't the final answer.

How can I remember this rule?

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Think "same base, add the powers"! When you multiply, you're combining groups of the same factor, so you add up how many times that factor appears total.

What happens with division of exponents?

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Division uses subtraction: a8÷a3=a8−3=a5 a^8 ÷ a^3 = a^{8-3} = a^5 . Remember: multiplication = addition, division = subtraction for exponent rules.

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