Multiply Powers with Same Base: Calculate 8²⋅8³⋅8⁵

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

When you multiply powers with the same base, you're essentially combining repeated multiplication. For example, $ 8^2 \cdot 8^3 = (8 \cdot 8) \cdot (8 \cdot 8 \cdot 8) = 8^5 $ - you're adding up all the 8's being multiplied together!

Q: What happens if the bases are different?

You cannot use the exponent addition rule! For example, $ 8^2 \cdot 3^2 $ must be calculated as $ 64 \cdot 9 = 576 $. The bases must be exactly the same to add exponents.

Q: How do I remember which operation to use with exponents?

Use this memory trick: Multiplying powers = Add exponents, Dividing powers = Subtract exponents, Power of a power = Multiply exponents. Practice with small numbers first!

Q: Can I use this rule with negative or fractional exponents?

Yes! The rule $ a^m \cdot a^n = a^{m+n} $ works for any exponents - positive, negative, fractions, or decimals. Just add them like regular numbers.

Exponent Addition with Multiple Terms

$8^2\cdot8^3\cdot8^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Alright, let's keep things simple.

00:09 We'll use the formula for multiplying powers.

00:13 When you multiply A to the power of M with A to the power of N,

00:19 the result is A to the power of M plus N.

00:23 Let's apply this formula in our exercise.

00:27 We'll connect the powers step by step.

00:36 And there you have it, the solution to our problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$8^2\cdot8^3\cdot8^5=$

Step-by-step solution

All bases are equal and therefore the exponents can be added together.

$8^2\cdot8^3\cdot8^5=8^{10}$

Final Answer

$8^{10}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $8^2 \cdot 8^3 \cdot 8^5 = 8^{2+3+5} = 8^{10}$
Check: Verify exponents sum correctly: 2 + 3 + 5 = 10 ✓

Common Mistakes

Avoid these frequent errors

Multiplying or adding the bases instead of adding exponents
Don't multiply 8 × 8 × 8 = 512 or add 8 + 8 + 8 = 24! This ignores the exponent rule and gives completely wrong results. Always keep the base unchanged and add only the exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add the exponents instead of multiplying them?

When you multiply powers with the same base, you're essentially combining repeated multiplication. For example, $8^2 \cdot 8^3 = (8 \cdot 8) \cdot (8 \cdot 8 \cdot 8) = 8^5$ - you're adding up all the 8's being multiplied together!

What happens if the bases are different?

You cannot use the exponent addition rule! For example, $8^2 \cdot 3^2$ must be calculated as $64 \cdot 9 = 576$ . The bases must be exactly the same to add exponents.

How do I remember which operation to use with exponents?

Use this memory trick: Multiplying powers = Add exponents, Dividing powers = Subtract exponents, Power of a power = Multiply exponents. Practice with small numbers first!

Can I use this rule with negative or fractional exponents?

Yes! The rule $a^m \cdot a^n = a^{m+n}$ works for any exponents - positive, negative, fractions, or decimals. Just add them like regular numbers.

What if I have more than three terms to multiply?

No problem! Keep adding all the exponents together. For example: $8^1 \cdot 8^2 \cdot 8^3 \cdot 8^4 = 8^{1+2+3+4} = 8^{10}$ . The rule works for any number of terms!

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