Simplify 6^2 × 6^8: Product Rule of Exponents Practice

Q: Why do we add exponents instead of multiplying them?

Think of $ 6^2 \times 6^8 $ as (6 × 6) × (6 × 6 × 6 × 6 × 6 × 6 × 6 × 6). You're multiplying 6 by itself 10 times total, so the answer is $ 6^{10} $!

Q: What if the bases are different numbers?

The product rule only works with the same base! For example, $ 6^2 \times 5^8 $ cannot be simplified using this rule because 6 and 5 are different bases.

Q: How is this different from the power rule?

The product rule adds exponents: $ 6^2 \times 6^8 = 6^{10} $. The power rule multiplies exponents: $ (6^2)^8 = 6^{16} $. Notice the parentheses make a big difference!

Q: Can I just calculate 6² and 6⁸ separately then multiply?

You could, but that's much harder! $ 6^2 = 36 $ and $ 6^8 = 1,679,616 $, so you'd multiply huge numbers. Using the rule gives $ 6^{10} $ immediately!

Q: What does 6¹⁰ actually equal?

While $ 6^{10} $ is the simplified form, if you need the actual number, it equals 60,466,176. But usually, leaving it as $ 6^{10} $ is the preferred answer!

Exponent Rules with Same Base Multiplication

Simplify the following equation:

$6^2\times6^8=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem together.

00:09 When we multiply powers with the same base, A, we add the exponents.

00:15 So, it's A raised to the power of N plus M.

00:19 Now, let's apply this rule to our exercise.

00:23 We add the exponents and use this sum as the new exponent.

00:27 And that's how we find the solution. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$6^2\times6^8=$

Step-by-step solution

To simplify the expression given, $6^2 \times 6^8$ , we will use the property of exponents which states that the product of two powers with the same base is the base raised to the sum of the exponents.

Let's apply the rule:

The base in both powers is $6$ .
The exponents are $2$ and $8$ .
According to the rule $a^m \times a^n = a^{m+n}$ , we add the exponents; therefore, $6^2 \times 6^8 = 6^{2+8}$ .
Simplifying further, this becomes $6^{10}$ .

Therefore, the simplified expression is $6^{10}$ .

The solution to the given problem is $6^{2+8}$ .

Final Answer

$6^{2+8}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: $6^2 \times 6^8 = 6^{2+8} = 6^{10}$
Check: Verify base stays same and exponents add: 2 + 8 = 10 ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply 2 × 8 = 16 to get $6^{16}$ ! This confuses the product rule with the power rule and gives a much larger incorrect answer. Always add exponents when multiplying same bases: $6^2 \times 6^8 = 6^{2+8} = 6^{10}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents instead of multiplying them?

Think of $6^2 \times 6^8$ as (6 × 6) × (6 × 6 × 6 × 6 × 6 × 6 × 6 × 6). You're multiplying 6 by itself 10 times total, so the answer is $6^{10}$ !

What if the bases are different numbers?

The product rule only works with the same base! For example, $6^2 \times 5^8$ cannot be simplified using this rule because 6 and 5 are different bases.

How is this different from the power rule?

The product rule adds exponents: $6^2 \times 6^8 = 6^{10}$ . The power rule multiplies exponents: $(6^2)^8 = 6^{16}$ . Notice the parentheses make a big difference!

Can I just calculate 6² and 6⁸ separately then multiply?

You could, but that's much harder! $6^2 = 36$ and $6^8 = 1,679,616$ , so you'd multiply huge numbers. Using the rule gives $6^{10}$ immediately!

What does 6¹⁰ actually equal?

While $6^{10}$ is the simplified form, if you need the actual number, it equals 60,466,176. But usually, leaving it as $6^{10}$ is the preferred answer!

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