Compare Powers: Evaluating 2³×2⁴×2⁸ and 2²×2³×2¹⁰

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

The multiplication rule for exponents states that $ a^m \times a^n = a^{m+n} $. Think of it this way: $ 2^3 \times 2^4 $ means (2×2×2) × (2×2×2×2) = 2 multiplied 7 times = $ 2^7 $.

Q: What if the bases were different numbers?

If the bases are different (like $ 2^3 \times 3^4 $), you cannot combine the exponents. The rule only works when the bases are exactly the same.

Q: How do I remember when to add vs multiply exponents?

Multiplying powers: add exponents ($ a^m \times a^n = a^{m+n} $)Power of a power: multiply exponents ($ (a^m)^n = a^{m \times n} $)

Q: Can I just calculate the actual values instead?

You could, but $ 2^{15} = 32,768 $ - that's a lot of calculation! Using exponent rules is faster and less error-prone than computing large numbers.

Q: What does the box symbol mean in the problem?

The box (□) is a placeholder where you need to insert the correct comparison symbol: $ > $ (greater than), $ (less than), or \( = $ (equal to).

Q: How can I double-check my work?

After simplifying both sides, make sure your exponent addition is correct: Left side: 3 + 4 + 8 = 15, Right side: 2 + 3 + 10 = 15. Since $ 2^{15} = 2^{15} $, the answer is =.

Exponent Laws with Multiple Products

Insert the compatible sign:

$>,<,=$

$2^3\times2^4\times2^8\Box2^2\times2^3\times2^{10}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Select the appropriate sign

00:03 Let's simplify each side using the formula for the multiplication of powers

00:06 We'll apply this formula to our exercise, each time for one multiplication

00:12 We'll maintain the base and add together the exponents

00:21 We'll apply the same formula in order to simplify the right side

00:31 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the compatible sign:

$>,<,=$

$2^3\times2^4\times2^8\Box2^2\times2^3\times2^{10}$

Step-by-step solution

To solve this problem, we'll apply the multiplication of powers rule, which states that for the same base $a$ , $a^m \times a^n = a^{m+n}$ . Let's simplify each expression:

For the left side:

$2^3 \times 2^4 \times 2^8$
Add the exponents since the bases are the same: $3 + 4 + 8 = 15$
The simplified form is $2^{15}$

For the right side:

$2^2 \times 2^3 \times 2^{10}$
Add the exponents as well: $2 + 3 + 10 = 15$
The simplified form is $2^{15}$

Now, comparing the two sides: $2^{15}$ and $2^{15}$ .

Since both are the same power of 2, we conclude that:

The correct sign to insert is $=$ .

Therefore, the solution to the problem is $\boxed{=}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: Add exponents: $3 + 4 + 8 = 15$ and $2 + 3 + 10 = 15$
Check: Both sides simplify to $2^{15}$ , so they are equal ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like $3 \times 4 \times 8 = 96$ = completely wrong result! This confuses the multiplication rule with the power rule. Always add 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?

The multiplication rule for exponents states that $a^m \times a^n = a^{m+n}$ . Think of it this way: $2^3 \times 2^4$ means (2×2×2) × (2×2×2×2) = 2 multiplied 7 times = $2^7$ .

What if the bases were different numbers?

If the bases are different (like $2^3 \times 3^4$ ), you cannot combine the exponents. The rule only works when the bases are exactly the same.

How do I remember when to add vs multiply exponents?

Multiplying powers: add exponents ( $a^m \times a^n = a^{m+n}$ )
Power of a power: multiply exponents ( $(a^m)^n = a^{m \times n}$ )

Can I just calculate the actual values instead?

You could, but $2^{15} = 32,768$ - that's a lot of calculation! Using exponent rules is faster and less error-prone than computing large numbers.

What does the box symbol mean in the problem?

The box (□) is a placeholder where you need to insert the correct comparison symbol: $>$ (greater than), $<$ (less than), or $=$ (equal to).

How can I double-check my work?

After simplifying both sides, make sure your exponent addition is correct: Left side: 3 + 4 + 8 = 15, Right side: 2 + 3 + 10 = 15. Since $2^{15} = 2^{15}$ , the answer is =.

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