Simplify the Expression: 4a^2 * a^4 * a^5 * a^3 + 20a^7

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

The product rule says $ a^m \times a^n = a^{m+n} $. Think of it as: $ a^2 \times a^3 = (a \times a) \times (a \times a \times a) = a^5 $. You're counting how many a's you have total!

Q: How do I know what the greatest common factor is?

Look for the largest number and lowest power of variables that divide both terms. Here: 4 divides both 4 and 20, and $ a^7 $ is the lowest power in $ a^{14} $ and $ a^7 $.

Q: Can I leave my answer as 4a^14 + 20a^7?

While technically correct, factored form is preferred because it's simpler and shows the relationship between terms. $ 4a^7(a^7 + 5) $ is much cleaner than the expanded form!

Q: What if the terms don't have a common factor?

If there's no common factor, then the expression is already in simplest form. But always check carefully - there might be a common factor you missed!

Q: Do I need to factor out the coefficient and variable separately?

Factor them together! The GCF of $ 4a^{14} $ and $ 20a^7 $ is $ 4a^7 $ (not just 4 or just $ a^7 $). Always look for the complete common factor.

Algebraic Factoring with Polynomial Terms

$4a^2\times a^4\times a^5\times a^3+20a^7=$

Simplify the expression as much as possible.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's make this problem simpler, step by step!

00:16 We'll start by solving the product of numbers.

00:22 When you multiply powers with the same base,

00:26 Add the exponents together, like this: A to the power of M plus N.

00:32 So, let's use this rule and add the powers in our problem.

00:49 Now, we factor out the common term from the brackets.

01:00 Great job! That's how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$4a^2\times a^4\times a^5\times a^3+20a^7=$

Simplify the expression as much as possible.

Step-by-step solution

In order to simplify the expression $4a^2 \times a^4 \times a^5 \times a^3 + 20a^7$ , we'll follow these steps:

Step 1: Combine the powers of $a$ in the first term using the product of powers rule:
$4a^2 \times a^4 \times a^5 \times a^3 = 4a^{2+4+5+3} = 4a^{14}$ .
Step 2: Rewrite the given expression with the simplified first term:
$4a^{14} + 20a^7$ .
Step 3: Identify and factor out the greatest common factor (GCF) from the expression:
Both terms have $4a^7$ as a common factor. Factor this out:
$4a^{14} + 20a^7 = 4a^7(a^7) + 4a^7(5) = 4a^7(a^7 + 5)$ .

Therefore, the simplified form of the expression is $4a^7(a^7 + 5)$ .

Final Answer

$4a^7(a^7+5)$

Key Points to Remember

Essential concepts to master this topic

Product Rule: Add exponents when multiplying same bases: $a^m \times a^n = a^{m+n}$
Technique: Find GCF of coefficients and variables: $4a^{14} + 20a^7 = 4a^7(a^7 + 5)$
Check: Distribute factored form back: $4a^7 \times a^7 + 4a^7 \times 5 = 4a^{14} + 20a^7$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to add exponents when multiplying
Don't multiply exponents like $a^2 \times a^4 = a^8$ = wrong answer! This gives $4a^8 + 20a^7$ which can't factor nicely. Always add exponents: $a^2 \times a^4 = a^{2+4} = a^6$ .

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?

The product rule says $a^m \times a^n = a^{m+n}$ . Think of it as: $a^2 \times a^3 = (a \times a) \times (a \times a \times a) = a^5$ . You're counting how many a's you have total!

How do I know what the greatest common factor is?

Look for the largest number and lowest power of variables that divide both terms. Here: 4 divides both 4 and 20, and $a^7$ is the lowest power in $a^{14}$ and $a^7$ .

Can I leave my answer as 4a^14 + 20a^7?

While technically correct, factored form is preferred because it's simpler and shows the relationship between terms. $4a^7(a^7 + 5)$ is much cleaner than the expanded form!

What if the terms don't have a common factor?

If there's no common factor, then the expression is already in simplest form. But always check carefully - there might be a common factor you missed!

Do I need to factor out the coefficient and variable separately?

Factor them together! The GCF of $4a^{14}$ and $20a^7$ is $4a^7$ (not just 4 or just $a^7$ ). Always look for the complete common factor.

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