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

$112^0=\text{?}$

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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