Simplify and Solve: 2^2·2^-3·2^4 / 2^3·2^-2·2^5

Q: Why do I add exponents when multiplying powers?

The multiplication rule for exponents states that $ a^m \cdot a^n = a^{m+n} $. This works because you're combining repeated multiplication: $ 2^2 \cdot 2^3 = (2 \cdot 2) \cdot (2 \cdot 2 \cdot 2) = 2^5 $.

Q: How do I handle negative exponents when adding?

Treat negative exponents like negative numbers in addition. For example: $ 2 + (-3) + 4 = 2 - 3 + 4 = 3 $. Don't change the operation - just add algebraically!

Q: Can I simplify each expression step by step?

Absolutely! Work left to right: first combine $ 2^2 \cdot 2^{-3} = 2^{-1} $, then multiply by $ 2^4 $ to get $ 2^{-1+4} = 2^3 $. This prevents errors with multiple exponents.

Q: Do I need to calculate the actual values to compare?

Not always! Since the base is the same (2), you can compare exponents directly: \( 2^3 because 3 8 confirms your answer.

Q: What if the bases were different?

With different bases like $ 3^2 $ vs $ 2^4 $, you must calculate the actual values: 9 vs 16. You cannot use exponent rules across different bases!

Exponent Rules with Expression Comparison

$2^2\cdot2^{-3}\cdot2^4\text{ }_{—\text{ }}2^3\cdot2^{-2}\cdot2^5$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the appropriate the sign

00:03 Let's simplify the left side

00:10 When multiplying powers with equal bases

00:13 The power of the result equals the sum of the powers

00:19 We'll apply this formula to our exercise, let's add up the powers

00:24 This is simplifying the left side

00:29 Let's simplify the right side

00:34 We'll apply the same formula again and add up the powers

00:42 Let's identify where the power is greater

00:47 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$2^2\cdot2^{-3}\cdot2^4\text{ }_{—\text{ }}2^3\cdot2^{-2}\cdot2^5$

Step-by-step solution

We start by simplifying each expression using the laws of exponents.

For the first expression $2^2 \cdot 2^{-3} \cdot 2^4$ :

Apply the multiplication of powers rule: $2^2 \cdot 2^{-3} = 2^{2 + (-3)} = 2^{-1}$ .
Now, multiply by $2^4$ : $2^{-1} \cdot 2^4 = 2^{-1 + 4} = 2^3$ .

Thus, the first expression simplifies to $2^3$ .

For the second expression $2^3 \cdot 2^{-2} \cdot 2^5$ :

Apply the multiplication of powers rule: $2^3 \cdot 2^{-2} = 2^{3 + (-2)} = 2^1$ .
Now, multiply by $2^5$ : $2^1 \cdot 2^5 = 2^{1 + 5} = 2^6$ .

Thus, the second expression simplifies to $2^6$ .

To compare $2^3$ and $2^6$ , we recognize that $2^6$ is greater than $2^3$ . Hence, the second expression is greater.

Thus, the correct answer is: $<$ .

Final Answer

$<$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $2^2 \cdot 2^{-3} \cdot 2^4 = 2^{2+(-3)+4} = 2^3$
Check: Calculate final values: $2^3 = 8$ and $2^6 = 64$ , so 8 < 64 ✓

Common Mistakes

Avoid these frequent errors

Adding exponents incorrectly with negative signs
Don't treat negative exponents like subtraction = wrong final power! Students often calculate 2 + (-3) + 4 as 2 - 3 - 4 = -5 instead of 2 + (-3) + 4 = 3. Always use proper integer addition rules with negative numbers.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add exponents when multiplying powers?

The multiplication rule for exponents states that $a^m \cdot a^n = a^{m+n}$ . This works because you're combining repeated multiplication: $2^2 \cdot 2^3 = (2 \cdot 2) \cdot (2 \cdot 2 \cdot 2) = 2^5$ .

How do I handle negative exponents when adding?

Treat negative exponents like negative numbers in addition. For example: $2 + (-3) + 4 = 2 - 3 + 4 = 3$ . Don't change the operation - just add algebraically!

Can I simplify each expression step by step?

Absolutely! Work left to right: first combine $2^2 \cdot 2^{-3} = 2^{-1}$ , then multiply by $2^4$ to get $2^{-1+4} = 2^3$ . This prevents errors with multiple exponents.

Do I need to calculate the actual values to compare?

Not always! Since the base is the same (2), you can compare exponents directly: $2^3 < 2^6$ because 3 < 6. But calculating 8 < 64 confirms your answer.

What if the bases were different?

With different bases like $3^2$ vs $2^4$ , you must calculate the actual values: 9 vs 16. You cannot use exponent rules across different bases!

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Simplify and Solve: 2^2·2^-3·2^4 / 2^3·2^-2·2^5

Exponent Rules with Expression Comparison

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I add exponents when multiplying powers?

How do I handle negative exponents when adding?

Can I simplify each expression step by step?

Do I need to calculate the actual values to compare?

What if the bases were different?

🌟 Unlock Your Math Potential

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