Simplify the Expression: 8³ × 8⁶ Using Laws of Exponents

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

Because exponents show how many times the base is multiplied by itself. $ 8^3 \times 8^6 $ means (8×8×8) × (8×8×8×8×8×8), which gives you 9 total eights = $ 8^9 $.

Q: What if the bases are different, like 2³ × 8⁶?

You cannot use the product rule! The bases must be exactly the same. For different bases, you'd need to calculate each part separately or find a common base first.

Q: How is this different from (8³)⁶?

That's the power rule! When raising a power to another power, you multiply exponents: $ (8^3)^6 = 8^{3 \times 6} = 8^{18} $. Don't confuse it with the product rule.

Q: Can I just calculate 8³ and 8⁶ separately first?

You could, but that's much harder! $ 8^3 = 512 $ and $ 8^6 = 262,144 $, so you'd multiply huge numbers. Using the product rule keeps everything simple as $ 8^9 $.

Q: What does 8⁹ actually equal?

$ 8^9 = 134,217,728 $ - that's over 134 million! The beauty of exponent rules is you can keep answers in exponential form without calculating these massive numbers.

Exponential Multiplication with Same Base

Simplify the following equation:

$8^3\times8^6=$

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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.

00:10 Remember, when multiplying exponents with the same base, like A,

00:15 we keep the base A and add the exponents, N plus M.

00:20 Let's use this rule to solve our exercise.

00:23 Add the exponents and raise the base to this power.

00:28 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$8^3\times8^6=$

Step-by-step solution

To solve this problem, we'll use the properties of exponents to simplify the expression:

Step 1: Identify the bases and exponents in the expression $8^3 \times 8^6$ .
Step 2: Apply the product of powers property, which states $a^m \times a^n = a^{m+n}$ when the bases are the same.
Step 3: Add the exponents: $3 + 6 = 9$ .

Now, let's work through these steps:

Step 1: Both terms, $8^3$ and $8^6$ , have the same base, 8.

Step 2: According to the product of powers property, we add the exponents: $8^{3+6}$ .

Step 3: Simplifying the exponents gives us $8^9$ .

Therefore, the simplified expression is $8^9$ .

Final Answer

$8^9$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: $8^3 \times 8^6 = 8^{3+6} = 8^9$
Check: Count total factors: 3 eights + 6 eights = 9 eights ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 3 × 6 = 18 to get $8^{18}$ ! This confuses the product rule with the power rule and gives a massively wrong answer. Always add exponents when multiplying same bases: 3 + 6 = 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?

Because exponents show how many times the base is multiplied by itself. $8^3 \times 8^6$ means (8×8×8) × (8×8×8×8×8×8), which gives you 9 total eights = $8^9$ .

What if the bases are different, like 2³ × 8⁶?

You cannot use the product rule! The bases must be exactly the same. For different bases, you'd need to calculate each part separately or find a common base first.

How is this different from (8³)⁶?

That's the power rule! When raising a power to another power, you multiply exponents: $(8^3)^6 = 8^{3 \times 6} = 8^{18}$ . Don't confuse it with the product rule.

Can I just calculate 8³ and 8⁶ separately first?

You could, but that's much harder! $8^3 = 512$ and $8^6 = 262,144$ , so you'd multiply huge numbers. Using the product rule keeps everything simple as $8^9$ .

What does 8⁹ actually equal?

$8^9 = 134,217,728$ - that's over 134 million! The beauty of exponent rules is you can keep answers in exponential form without calculating these massive numbers.

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