Complete the Expression: 4³ × x³ × 6³ Multiplication Problem

Q: Why can I combine these terms when they have different bases?

You can combine them because they all have the same exponent (3)! The power-of-a-product rule says $ a^n \times b^n \times c^n = (a \times b \times c)^n $. The bases can be different, but the exponents must match.

Q: What if the exponents were different, like 4² × x³ × 6⁴?

Then you cannot combine them using this rule! The power-of-a-product rule only works when all exponents are identical. Different exponents mean you'd have to calculate each term separately.

Q: Do I multiply 4 × 6 = 24 first inside the parentheses?

No! Leave it as $ (4 \times x \times 6)^3 $. The question asks for the equivalent expression, not the simplified numerical result. Keep the variable x separate.

Q: How is this different from adding exponents?

Adding exponents happens when you have the same base: $ x^2 \times x^3 = x^5 $. Here we have different bases with the same exponent, so we use the product-to-power rule instead.

Q: Can I write it as (24x)³ instead?

Mathematically yes, but the answer choices show $ (4 \times x \times 6)^3 $. In multiple choice questions, match the exact format of the given options rather than simplifying further.

Exponent Rules with Product-to-Power Conversion

Insert the corresponding expression:

$4^3\times x^3\times6^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:12 Imagine a product, where each part is raised to its own power, N.

00:17 This can be changed into the entire product in parentheses, raised to power N.

00:22 Now, we will apply this formula to solve our exercise.

00:27 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$4^3\times x^3\times6^3=$

Step-by-step solution

To solve the problem, let's use the power of a product rule, which states that the product of several terms raised to the same power can be expressed as one parenthesis where the terms are multiplied, raised to that power.

Given the expression:

$4^3 \times x^3 \times 6^3$

We observe that all the terms $4$ , $x$ , and $6$ are each raised to the power of $3$ . Therefore, we can represent this expression using a single power as follows:

$\left(4 \times x \times 6\right)^3$

This transformation uses the formula $(a \times b \times c)^n = a^n \times b^n \times c^n$ , taking advantage of corresponding exponents.

Therefore, the simplified expression is $\left(4 \times x \times 6\right)^3$ .

Final Answer

$\left(4\times x\times6\right)^3$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Same exponents on products combine: $a^n \times b^n = (a \times b)^n$
Technique: Group bases first: $4^3 \times x^3 \times 6^3 = (4 \times x \times 6)^3$
Check: Expand backwards: $(4 \times x \times 6)^3 = 4^3 \times x^3 \times 6^3$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of keeping them the same
Don't change 3 + 3 + 3 = 9 and write $(4 \times x \times 6)^9$ ! This confuses multiplication rules with addition. The exponents stay the same because we're using the product-to-power rule. Always keep the original exponent when combining equal powers.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine these terms when they have different bases?

You can combine them because they all have the same exponent (3)! The power-of-a-product rule says $a^n \times b^n \times c^n = (a \times b \times c)^n$ . The bases can be different, but the exponents must match.

What if the exponents were different, like 4² × x³ × 6⁴?

Then you cannot combine them using this rule! The power-of-a-product rule only works when all exponents are identical. Different exponents mean you'd have to calculate each term separately.

Do I multiply 4 × 6 = 24 first inside the parentheses?

No! Leave it as $(4 \times x \times 6)^3$ . The question asks for the equivalent expression, not the simplified numerical result. Keep the variable x separate.

How is this different from adding exponents?

Adding exponents happens when you have the same base: $x^2 \times x^3 = x^5$ . Here we have different bases with the same exponent, so we use the product-to-power rule instead.

Can I write it as (24x)³ instead?

Mathematically yes, but the answer choices show $(4 \times x \times 6)^3$ . In multiple choice questions, match the exact format of the given options rather than simplifying further.

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