Evaluate (2×3)² : Order of Operations with Square Numbers

Q: Why can't I just multiply 2×3 first to get 6²?

You can! $ (2 \times 3)^2 = 6^2 = 36 $ is correct. But the question asks for the equivalent expression using the power of a product rule, which means keeping the factors separate: $ 2^2 \times 3^2 $.

Q: What's the difference between (2×3)² and 2×3²?

Big difference! $ (2 \times 3)^2 $ means square the entire product (36), while $ 2 \times 3^2 $ means multiply 2 by 3² = 2 × 9 = 18. Parentheses change everything!

Q: Does this rule work with more than two factors?

Absolutely! For example: $ (2 \times 3 \times 4)^2 = 2^2 \times 3^2 \times 4^2 $. The exponent distributes to every factor inside the parentheses.

Q: What if the exponent is different, like cubed?

Same rule applies! $ (2 \times 3)^3 = 2^3 \times 3^3 $. No matter what the exponent is, it distributes to each factor in the product.

Q: How do I remember this rule?

Think of it as "sharing the power" - when you have a product raised to a power, each factor gets its own copy of that power. The parentheses act like a distribution center for the exponent!

Power of a Product with Exponent Rules

Insert the corresponding expression:

$\left(2\times3\right)^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 In order to open parentheses containing a multiplication operation with an outer exponent

00:06 We raise each factor to the power

00:10 We will apply this formula to our exercise

00:15 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(2\times3\right)^2=$

Step-by-step solution

The given expression is $\left(2\times3\right)^2$ . We need to apply the rule of exponents known as the "Power of a Product." This rule states that when you have a product raised to an exponent, you can apply the exponent to each factor in the product individually. Mathematically, this is expressed as: $(a \times b)^n = a^n \times b^n$ .

In this case, the expression $\left(2\times3\right)^2$ follows this rule with $a = 2$ and $b = 3$ , and $n = 2$ .

First, apply the exponent to the first factor: $2^2$ .
Next, apply the exponent to the second factor: $3^2$ .

Therefore, by applying the "Power of a Product" rule, the expression becomes: $2^2 \times 3^2$ .

Final Answer

$2^2\times3^2$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, distribute the exponent to each factor
Technique: Transform $(2 \times 3)^2$ into $2^2 \times 3^2$
Check: Both $(2 \times 3)^2 = 36$ and $2^2 \times 3^2 = 4 \times 9 = 36$ ✓

Common Mistakes

Avoid these frequent errors

Calculating the product first then squaring
Don't evaluate $2 \times 3 = 6$ first then write $6^2 \times 6^2$ = wrong answer! This doubles the exponent and gives 1296 instead of 36. Always distribute the exponent to each factor separately: $(a \times b)^n = a^n \times b^n$ .

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why can't I just multiply 2×3 first to get 6²?

You can! $(2 \times 3)^2 = 6^2 = 36$ is correct. But the question asks for the equivalent expression using the power of a product rule, which means keeping the factors separate: $2^2 \times 3^2$ .

What's the difference between (2×3)² and 2×3²?

Big difference! $(2 \times 3)^2$ means square the entire product (36), while $2 \times 3^2$ means multiply 2 by 3² = 2 × 9 = 18. Parentheses change everything!

Does this rule work with more than two factors?

Absolutely! For example: $(2 \times 3 \times 4)^2 = 2^2 \times 3^2 \times 4^2$ . The exponent distributes to every factor inside the parentheses.

What if the exponent is different, like cubed?

Same rule applies! $(2 \times 3)^3 = 2^3 \times 3^3$ . No matter what the exponent is, it distributes to each factor in the product.

How do I remember this rule?

Think of it as "sharing the power" - when you have a product raised to a power, each factor gets its own copy of that power. The parentheses act like a distribution center for the exponent!

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