Calculate (2×6)³: Evaluating the Cube of a Product

Q: Why can't I just multiply 2×6×3 to get the answer?

The exponent 3 means multiply the entire product by itself 3 times, not multiply by 3. So $ (2×6)^3 = (2×6)×(2×6)×(2×6) $, which equals $ 12×12×12 = 1728 $.

Q: Do I have to use the power of a product rule?

You have two valid approaches: either use $ (2×6)^3 = 2^3×6^3 $ or simplify to $ 12^3 $ first. Both give the same answer, but the power rule helps you identify equivalent expressions.

Q: What if there are more than two numbers inside the parentheses?

The rule works the same way! For $ (2×3×5)^4 $, you get $ 2^4×3^4×5^4 $. Just apply the exponent to each factor individually.

Q: How do I know which answer choice is correct?

Look for the expression that applies the exponent 3 to both factors. The correct answer $ 2^3×6^3 $ shows each base raised to the power of 3, following the power of a product rule.

Q: Can I calculate the numerical value to check my work?

Absolutely! Calculate both expressions: $ (2×6)^3 = 12^3 = 1728 $ and $ 2^3×6^3 = 8×216 = 1728 $. Getting the same result confirms you've applied the rule correctly!

Power of Products with Exponent Rules

Choose the expression that corresponds to the following:

$\left(2\times6\right)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this expression.

00:12 When you see parentheses with an exponent over multiplication,

00:16 raise each number inside to that power.

00:20 We'll use this formula to solve our exercise.

00:26 Here's how it's done!

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:

$\left(2\times6\right)^3=$

Step-by-step solution

We are given the expression $\left(2\times6\right)^3$ and need to simplify it using the power of a product rule in exponents.

The power of a product rule states that when you have a product inside a power, you can apply the exponent to each factor in the product individually. In mathematical terms, the rule is expressed as:

$(a \cdot b)^n = a^n \cdot b^n$

Applying this to our expression, we have:

$\left(2\times6\right)^3 = 2^3\times6^3$

This means that each term inside the parentheses is raised to the power of 3 separately.

Therefore, the expression $\left(2\times6\right)^3$ simplifies to $2^3\times6^3$ as per the power of a product rule.

Final Answer

$2^3\times6^3$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply the exponent to each factor separately
Technique: Transform $(2×6)^3$ into $2^3×6^3$
Check: Both methods give 1728: $12^3 = 8×216 = 1728$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the base by the exponent
Don't calculate $(2×6)^3$ as $12×3 = 36$ ! This treats the exponent as a multiplier instead of repeated multiplication. Always apply the power of a product rule: $(a×b)^n = a^n×b^n$ .

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 2×6×3 to get the answer?

The exponent 3 means multiply the entire product by itself 3 times, not multiply by 3. So $(2×6)^3 = (2×6)×(2×6)×(2×6)$ , which equals $12×12×12 = 1728$ .

Do I have to use the power of a product rule?

You have two valid approaches: either use $(2×6)^3 = 2^3×6^3$ or simplify to $12^3$ first. Both give the same answer, but the power rule helps you identify equivalent expressions.

What if there are more than two numbers inside the parentheses?

The rule works the same way! For $(2×3×5)^4$ , you get $2^4×3^4×5^4$ . Just apply the exponent to each factor individually.

How do I know which answer choice is correct?

Look for the expression that applies the exponent 3 to both factors. The correct answer $2^3×6^3$ shows each base raised to the power of 3, following the power of a product rule.

Can I calculate the numerical value to check my work?

Absolutely! Calculate both expressions: $(2×6)^3 = 12^3 = 1728$ and $2^3×6^3 = 8×216 = 1728$ . Getting the same result confirms you've applied the rule correctly!

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