Simplify the Expression: 6^a × 6^4a Using Laws of Exponents

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

The multiplication rule says $ b^m \times b^n = b^{m+n} $. Think of it as: $ 6^a $ means 6 multiplied by itself 'a' times, and $ 6^{4a} $ means 6 multiplied by itself '4a' times. Together, that's a + 4a = 5a times total!

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

The power rule $ (b^m)^n = b^{mn} $ is for raising a power to another power - you multiply exponents. But here we're multiplying two powers with the same base, so we add exponents.

Q: Why are both answers B and C considered correct?

Answer B shows the intermediate step $ 6^{a+4a} $ and answer C shows the final simplified form $ 6^{5a} $. Both represent correct parts of the complete solution process.

Q: Can I check my answer somehow?

Yes! Try substituting a simple value like a = 1. Then $ 6^1 \times 6^4 = 6 \times 1296 = 7776 $, and $ 6^{5(1)} = 6^5 = 7776 $. They match! ✓

Exponent Multiplication with Same Base

Reduce the following equation:

$6^a\times6^{4a}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this problem!

00:13 In math, when you multiply two numbers with the same base, A,

00:17 you keep the base and add the exponents, N plus M.

00:22 So, let's use this rule to solve our problem.

00:26 Keep the base the same, and add the exponents together.

00:31 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$6^a\times6^{4a}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Identify the given expression and its components.
Recognize that both terms share the same base.
Apply the rule for multiplying powers with the same base, $b^m \times b^n = b^{m+n}$ .
Calculate the exponent by adding the powers together.

Let's work through these steps:

Given the expression:
$6^a \times 6^{4a}$

Since both terms share the same base (6), we use the rule for multiplying powers with the same base:

$6^a \times 6^{4a} = 6^{a + 4a}$ .

Simplify the exponent:

$a + 4a = 5a$ .

Thus, the expression simplifies to:

$6^{5a}$ .

Since the correct placement of our steps has directly given us choice C and corroborated choice B as intermediary work, B+C represents the comprehensive workings of the problem; thus, B+C are correct is the most comprehensive solution.

Final Answer

B+C are correct

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add exponents: $b^m \times b^n = b^{m+n}$
Technique: Add a + 4a = 5a to get $6^{5a}$
Check: Verify by expanding: $6^a \times 6^{4a} = 6^{a+4a} = 6^{5a}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply a × 4a = 4a² to get $6^{4a²}$ ! This confuses the multiplication rule with the power rule. Always add exponents when multiplying same bases: a + 4a = 5a.

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 multiplication rule says $b^m \times b^n = b^{m+n}$ . Think of it as: $6^a$ means 6 multiplied by itself 'a' times, and $6^{4a}$ means 6 multiplied by itself '4a' times. Together, that's a + 4a = 5a times total!

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

The power rule $(b^m)^n = b^{mn}$ is for raising a power to another power - you multiply exponents. But here we're multiplying two powers with the same base, so we add exponents.

How do I simplify a + 4a?

Think of it like combining like terms: a + 4a = 1a + 4a = 5a. It's just like saying 1 apple + 4 apples = 5 apples!

Why are both answers B and C considered correct?

Answer B shows the intermediate step $6^{a+4a}$ and answer C shows the final simplified form $6^{5a}$ . Both represent correct parts of the complete solution process.

Can I check my answer somehow?

Yes! Try substituting a simple value like a = 1. Then $6^1 \times 6^4 = 6 \times 1296 = 7776$ , and $6^{5(1)} = 6^5 = 7776$ . They match! ✓

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