Calculate (2×5×6×4)^20: Solving a Complex Power Expression

Q: Why are both 240^20 and 10^20×24^20 correct answers?

Both expressions are mathematically equivalent! Since $ 240 = 10 \times 24 $, we have $ 240^{20} = (10 \times 24)^{20} = 10^{20} \times 24^{20} $ by the power of a product rule.

Q: Should I multiply out 2×5×6×4 first or group the factors?

Both approaches work! You can calculate $ 2 \times 5 \times 6 \times 4 = 240 $ directly, or group strategically like $ (2 \times 5) \times (6 \times 4) = 10 \times 24 $. The grouping method reveals more equivalent expressions.

Q: What's the power of a product rule exactly?

The rule states: $ (a \times b)^n = a^n \times b^n $. When you raise a product to a power, you can distribute that power to each factor separately.

Q: How can I check if two exponential expressions are equal?

Use the properties of exponents! Check if the bases multiply to the same value and have the same total power. For example: $ 240^{20} $ and $ 10^{20} \times 24^{20} $ both equal $ (10 \times 24)^{20} $.

Power of Products with Multiple Factors

Insert the corresponding expression:

$\left(2\times5\times6\times4\right)^{20}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 Let's solve one multiplication at a time

00:15 This is one solution to the question

00:21 According to the laws of exponents, a multiplication where all terms are raised to power (N)

00:25 equals a multiplication where each factor is raised to the same power (N)

00:28 We'll apply this formula to our exercise

00:35 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(2\times5\times6\times4\right)^{20}=$

Step-by-step solution

To solve the problem, we need to evaluate the expression $\left(2 \times 5 \times 6 \times 4\right)^{20}$ .

Step 1: Calculate the product inside the parenthesis:

$2 \times 5 \times 6 \times 4$ .

Breaking it down, we have:

$2 \times 5 = 10$ .
$6 \times 4 = 24$ .
Therefore, the product is $10 \times 24$ .

Step 2: Apply the power of a product rule:

$(10 \times 24)^{20}$ can be rewritten using the power of a product rule:

$10^{20} \times 24^{20}$ .

Step 3: Compare with the given choices:

Choice 2: $10^{20} \times 24^{20}$ matches the calculation directly.
Choice 3: $240^{20}$ matches $(10 \times 24)^{20}$ if calculated directly without breaking into components.

Therefore, both expressions in options Choice 2 and Choice 3 are equivalent to the given expression $(2 \times 5 \times 6 \times 4)^{20}$ .

Thus, the correct choice according to the problem is: B+C are correct.

Final Answer

B+C are correct

Key Points to Remember

Essential concepts to master this topic

Power Rule: $(a \times b)^n = a^n \times b^n$ distributes over multiplication
Technique: Group factors first: $2 \times 5 = 10$ and $6 \times 4 = 24$
Check: Verify $240^{20} = (10 \times 24)^{20} = 10^{20} \times 24^{20}$ ✓

Common Mistakes

Avoid these frequent errors

Computing the full product before applying power rules
Don't calculate 2×5×6×4 = 240 and then write only 240^20! This misses equivalent forms that use power rules. The question asks for corresponding expressions, so you need both 240^20 AND 10^20×24^20. Always recognize multiple valid representations using exponent properties.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are both 240^20 and 10^20×24^20 correct answers?

Both expressions are mathematically equivalent! Since $240 = 10 \times 24$ , we have $240^{20} = (10 \times 24)^{20} = 10^{20} \times 24^{20}$ by the power of a product rule.

Should I multiply out 2×5×6×4 first or group the factors?

Both approaches work! You can calculate $2 \times 5 \times 6 \times 4 = 240$ directly, or group strategically like $(2 \times 5) \times (6 \times 4) = 10 \times 24$ . The grouping method reveals more equivalent expressions.

What's the power of a product rule exactly?

The rule states: $(a \times b)^n = a^n \times b^n$ . When you raise a product to a power, you can distribute that power to each factor separately.

Why isn't 10×6^20×4^20 a correct answer?

This expression doesn't follow proper exponent rules! The original expression is $(2 \times 5 \times 6 \times 4)^{20}$ , so every factor should be raised to the 20th power, not just some of them.

How can I check if two exponential expressions are equal?

Use the properties of exponents! Check if the bases multiply to the same value and have the same total power. For example: $240^{20}$ and $10^{20} \times 24^{20}$ both equal $(10 \times 24)^{20}$ .

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