Calculate (5×7)³: Evaluating a Cubed Expression

Q: Why can't I just cube one number and multiply by the other?

Because the exponent 3 applies to the entire product $ (5 \times 7) $. Think of it as $ (5 \times 7) \times (5 \times 7) \times (5 \times 7) $, which gives you three 5's and three 7's multiplied together.

Q: Can I multiply 5×7 first, then cube the result?

Yes! Both methods work: $ (5 \times 7)^3 = 35^3 = 42,875 $ or $ 5^3 \times 7^3 = 125 \times 343 = 42,875 $. The question asks for the equivalent expression, which is $ 5^3 \times 7^3 $.

Q: What's wrong with the answer 35³ × 35³?

$ 35^3 \times 35^3 = 35^6 $, not $ 35^3 $! This would be squaring the cubed result, giving you a much larger number than what we want.

Q: Does this rule work for any exponent?

Absolutely! The power of a product rule $ (a \times b)^n = a^n \times b^n $ works for any exponent n, whether it's 2, 3, 10, or even fractions and negative numbers.

Q: How do I remember when to use this rule?

Look for parentheses around a product with an exponent outside. The key phrase is "power of a product" - when you see this pattern, distribute the exponent to each factor inside the parentheses.

Power of Product Rule with Cubed Expressions

Choose the expression that corresponds to the following:

$\left(5\times7\right)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:03 In order to open parentheses with an exponent over multiplication

00:06 We raise each factor to the power

00:09 We'll use this formula in our exercise

00:13 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(5\times7\right)^3=$

Step-by-step solution

The problem requires us to simplify the expression $(5 \times 7)^3$ using the power of a product rule.

The power of a product rule states that for any numbers $a$ and $b$ , and any integer $n$ , the expression $(a \times b)^n$ can be expanded to $a^n \times b^n$ .

Applying this rule to the given expression:

Identify the values of $a$ and $b$ as $5$ and $7$ , respectively.
Identify $n$ as $3$ .
Substitute using the rule:
$(5 \times 7)^3 = 5^3 \times 7^3$

The simplified expression is therefore $5^3 \times 7^3$ .

Final Answer

$5^3\times7^3$

Key Points to Remember

Essential concepts to master this topic

Rule: For $(a \times b)^n$ , apply exponent to each factor separately
Technique: $(5 \times 7)^3 = 5^3 \times 7^3$ using power distribution
Check: Calculate both ways: $35^3 = 42,875$ and $5^3 \times 7^3 = 125 \times 343 = 42,875$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't calculate $(5 \times 7)^3$ as $5^3 \times 7 = 125 \times 7 = 875$ ! This ignores the power rule and gives a completely wrong answer (875 instead of 42,875). Always distribute the exponent to every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

Why can't I just cube one number and multiply by the other?

Because the exponent 3 applies to the entire product $(5 \times 7)$ . Think of it as $(5 \times 7) \times (5 \times 7) \times (5 \times 7)$ , which gives you three 5's and three 7's multiplied together.

Can I multiply 5×7 first, then cube the result?

Yes! Both methods work: $(5 \times 7)^3 = 35^3 = 42,875$ or $5^3 \times 7^3 = 125 \times 343 = 42,875$ . The question asks for the equivalent expression, which is $5^3 \times 7^3$ .

What's wrong with the answer 35³ × 35³?

$35^3 \times 35^3 = 35^6$ , not $35^3$ ! This would be squaring the cubed result, giving you a much larger number than what we want.

Does this rule work for any exponent?

Absolutely! The power of a product rule $(a \times b)^n = a^n \times b^n$ works for any exponent n, whether it's 2, 3, 10, or even fractions and negative numbers.

How do I remember when to use this rule?

Look for parentheses around a product with an exponent outside. The key phrase is "power of a product" - when you see this pattern, distribute the exponent to each factor inside the parentheses.

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