Calculate (2×7×5)³: Product of Numbers Raised to Third Power

Q: Why can't I just multiply 2×7×5 first and then cube it?

You absolutely can do that! $ (2 \times 7 \times 5)^3 = (70)^3 = 343,000 $. However, the question asks for the expanded form using the power of product rule: $ 2^3 \times 7^3 \times 5^3 $.

Q: How do I remember the power of product rule?

Think of it like sharing! When you have $ (abc)^3 $, the exponent 3 gets shared equally with each factor inside. So a gets cubed, b gets cubed, and c gets cubed.

Q: What if there are more than 3 factors inside the parentheses?

The rule works the same way no matter how many factors you have! For example: $ (abcd)^2 = a^2 \times b^2 \times c^2 \times d^2 $. Just distribute the exponent to every single factor.

Q: Do I need to calculate the actual values of 2³, 7³, and 5³?

Not unless the problem asks for it! The answer $ 2^3 \times 7^3 \times 5^3 $ is already in its correct expanded form. If you want the numerical answer: $ 8 \times 343 \times 125 = 343,000 $.

Q: What's the difference between (2×7×5)³ and 2×7×5³?

Huge difference! In $ (2 \times 7 \times 5)^3 $, the parentheses mean everything gets cubed. In $ 2 \times 7 \times 5^3 $, only the 5 gets cubed while 2 and 7 stay as they are.

Exponent Rules with Product Powers

$(2\times7\times5)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify!

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

00:12 When you have a multiplication in parentheses raised to power N,

00:17 You raise each number inside separately to the power of N.

00:22 Now, let's apply this formula in our exercise!

00:26 And that's how we solve the question. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(2\times7\times5)^3=$

Step-by-step solution

To solve the problem, we need to apply the power of a product exponent rule. This rule states that when you raise a product to a power, it's the same as raising each factor to that power. In mathematical terms, if you have $(abc)^n$ , it is equivalent to $a^n \times b^n \times c^n$ .

Let's apply this rule step by step:

Our original expression is: $(2 \times 7 \times 5)^3$ .

We first identify the factors inside the parentheses as $2$ , $7$ , and $5$ .

According to the Power of a Product rule, we can distribute the exponent $3$ to each factor:

First, raise $2$ to the power of $3$ to get $2^3$ .

Then, raise $7$ to the power of $3$ to get $7^3$ .

Finally, raise $5$ to the power of $3$ to get $5^3$ .

Therefore, the expression $(2 \times 7 \times 5)^3$ simplifies to $2^3 \times 7^3 \times 5^3$ .

Final Answer

$2^3\times7^3\times5^3$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(abc)^n = a^n \times b^n \times c^n$
Technique: Distribute exponent 3 to each factor: $2^3 \times 7^3 \times 5^3$
Check: Verify by calculating: $8 \times 343 \times 125 = 343,000$ ✓

Common Mistakes

Avoid these frequent errors

Adding the exponent instead of distributing it
Don't calculate $2 \times 7 \times 5 + 3 = 73$ ! This treats the exponent like addition and ignores the power rule completely. Always distribute the exponent to each factor inside the parentheses using $(abc)^n = a^n \times b^n \times c^n$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply 2×7×5 first and then cube it?

You absolutely can do that! $(2 \times 7 \times 5)^3 = (70)^3 = 343,000$ . However, the question asks for the expanded form using the power of product rule: $2^3 \times 7^3 \times 5^3$ .

How do I remember the power of product rule?

Think of it like sharing! When you have $(abc)^3$ , the exponent 3 gets shared equally with each factor inside. So a gets cubed, b gets cubed, and c gets cubed.

What if there are more than 3 factors inside the parentheses?

The rule works the same way no matter how many factors you have! For example: $(abcd)^2 = a^2 \times b^2 \times c^2 \times d^2$ . Just distribute the exponent to every single factor.

Do I need to calculate the actual values of 2³, 7³, and 5³?

Not unless the problem asks for it! The answer $2^3 \times 7^3 \times 5^3$ is already in its correct expanded form. If you want the numerical answer: $8 \times 343 \times 125 = 343,000$ .

What's the difference between (2×7×5)³ and 2×7×5³?

Huge difference! In $(2 \times 7 \times 5)^3$ , the parentheses mean everything gets cubed. In $2 \times 7 \times 5^3$ , only the 5 gets cubed while 2 and 7 stay as they are.

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Calculate (2×7×5)³: Product of Numbers Raised to Third Power

Exponent Rules with Product Powers

❤️ 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 can't I just multiply 2×7×5 first and then cube it?

How do I remember the power of product rule?

What if there are more than 3 factors inside the parentheses?

Do I need to calculate the actual values of 2³, 7³, and 5³?

What's the difference between (2×7×5)³ and 2×7×5³?

🌟 Unlock Your Math Potential

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