Calculate (9×2×5)³: Cube of a Three-Number Product

Q: Why can't I just multiply 9×2×5 first and then cube the result?

You absolutely can do that! $ 9\times2\times5 = 90 $, so $ 90^3 = 729,000 $. Both methods give the same answer: $ 729\times8\times125 = 729,000 $.

Q: How do I know which method is easier?

If the numbers inside multiply to a nice round number (like 90), calculate inside first. If the individual powers are easier to work with, use the power rule. Practice both!

Q: What if there are more than three numbers being multiplied?

The same rule applies! $ (a\times b\times c\times d)^n = a^n\times b^n\times c^n\times d^n $. Each factor gets raised to the power separately.

Q: Do I always have to calculate the final numerical answer?

Not always! Sometimes the question asks for the expression form like $ 9^3\times2^3\times5^3 $. Read carefully to see what format they want.

Q: What's the difference between this and (9+2+5)³?

Huge difference! With addition $ (9+2+5)^3 = 16^3 $, you must add first. With multiplication, you can distribute the exponent to each factor using the power rule.

Exponent Rules with Product Powers

$(9\times2\times5)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's begin!

00:07 We'll use the power rule for multiplication.

00:11 Any number multiplied and then raised to the power of N...

00:15 Means each number is separately raised to the power N.

00:19 We'll use this idea in our exercise.

00:22 Let's expand the brackets and raise each number to its power.

00:27 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(9\times2\times5)^3=$

Step-by-step solution

We use the law of exponents for a power that is applied to parentheses in which terms are multiplied:

$(x\cdot y)^n=x^n\cdot y^n$

We apply the rule to the problem:

$(9\cdot2\cdot5)^3=9^3\cdot2^3\cdot5^3$

When we apply the power within parentheses to the product of the terms we do so separately and maintain the multiplication,

Therefore, the correct answer is option B.

Final Answer

$9^3\times2^3\times5^3$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, apply exponent to each factor
Technique: $(9\times2\times5)^3 = 9^3\times2^3\times5^3$
Check: Calculate $9^3=729, 2^3=8, 5^3=125$ then multiply: 729×8×125 ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only one factor
Don't write $(9\times2\times5)^3 = 9^3\times2\times5$ = 729×2×5 = 7,290! This ignores the power rule and gives a completely wrong result. Always apply the exponent to every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ (2^3)^6 = $$

FAQ

Everything you need to know about this question

Why can't I just multiply 9×2×5 first and then cube the result?

You absolutely can do that! $9\times2\times5 = 90$ , so $90^3 = 729,000$ . Both methods give the same answer: $729\times8\times125 = 729,000$ .

How do I know which method is easier?

If the numbers inside multiply to a nice round number (like 90), calculate inside first. If the individual powers are easier to work with, use the power rule. Practice both!

What if there are more than three numbers being multiplied?

The same rule applies! $(a\times b\times c\times d)^n = a^n\times b^n\times c^n\times d^n$ . Each factor gets raised to the power separately.

Do I always have to calculate the final numerical answer?

Not always! Sometimes the question asks for the expression form like $9^3\times2^3\times5^3$ . Read carefully to see what format they want.

What's the difference between this and (9+2+5)³?

Huge difference! With addition $(9+2+5)^3 = 16^3$ , you must add first. With multiplication, you can distribute the exponent to each factor using the power rule.

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Calculate (9×2×5)³: Cube of a Three-Number Product

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 9×2×5 first and then cube the result?

How do I know which method is easier?

What if there are more than three numbers being multiplied?

Do I always have to calculate the final numerical answer?

What's the difference between this and (9+2+5)³?

🌟 Unlock Your Math Potential

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