Simplify the Expression: a^2 × a^3 × a^4 Using Exponent Rules

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

Think of what exponents mean! $ a^2 \times a^3 $ means (a·a) × (a·a·a), which gives you 5 total factors of a, so $ a^5 $. Adding exponents counts the total factors correctly.

Q: What if the bases are different, like a² × b³?

You cannot combine them! The exponent rule only works with the same base. Different bases like $ a^2 \times b^3 $ stay separate.

Q: Does this work with more than three terms?

Absolutely! You can add as many exponents as you want: $ a^1 \times a^2 \times a^3 \times a^4 \times a^5 = a^{1+2+3+4+5} = a^{15} $

Q: How can I remember not to multiply the exponents?

Try a simple example: $ 2^2 \times 2^3 = 4 \times 8 = 32 = 2^5 $. If you multiplied exponents, you'd get $ 2^6 = 64 $ - clearly wrong!

Q: What's the final simplified answer for this problem?

The expression $ a^2 \times a^3 \times a^4 $ simplifies to $ a^{2+3+4} = a^9 $. The correct choice shows the process $ a^{2+3+4} $.

Exponent Rules with Same Base Multiplication

Reduce the following equation:

$a^2\times a^3\times a^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 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)

00:11 We will apply this formula to our exercise

00:16 We'll maintain the base and add the exponents together

00:24 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:

$a^2\times a^3\times a^4=$

Step-by-step solution

To solve the problem of simplifying $a^2 \times a^3 \times a^4$ , we apply the exponent rule for multiplying powers with the same base.

This rule states that to multiply powers with the same base, we add their exponents:

$a^m \times a^n = a^{m+n}$

Applying this rule to the problem at hand:

The given expression is $a^2 \times a^3 \times a^4$
We recognize that all parts have the same base, so we can add the exponents together: $2 + 3 + 4$ .
Therefore, we simplify the expression to $a^{2+3+4}$ .

This matches with choice 4, $a^{2+3+4}$ .

Final Answer

$a^{2+3+4}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add exponents together
Technique: $a^2 \times a^3 \times a^4 = a^{2+3+4} = a^9$
Check: Count total factors: a·a·a·a·a·a·a·a·a equals nine a's ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply exponents like $a^{2\times3\times4} = a^{24}$ ! This gives a completely wrong answer with way too many factors. Always add exponents when bases are the same: $a^{2+3+4} = a^9$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

Think of what exponents mean! $a^2 \times a^3$ means (a·a) × (a·a·a), which gives you 5 total factors of a, so $a^5$ . Adding exponents counts the total factors correctly.

What if the bases are different, like a² × b³?

You cannot combine them! The exponent rule only works with the same base. Different bases like $a^2 \times b^3$ stay separate.

Does this work with more than three terms?

Absolutely! You can add as many exponents as you want: $a^1 \times a^2 \times a^3 \times a^4 \times a^5 = a^{1+2+3+4+5} = a^{15}$

How can I remember not to multiply the exponents?

Try a simple example: $2^2 \times 2^3 = 4 \times 8 = 32 = 2^5$ . If you multiplied exponents, you'd get $2^6 = 64$ - clearly wrong!

What's the final simplified answer for this problem?

The expression $a^2 \times a^3 \times a^4$ simplifies to $a^{2+3+4} = a^9$ . The correct choice shows the process $a^{2+3+4}$ .

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