Solve (c×b×a)²: Finding the Square of a Three-Variable Product

Q: Why do all variables get squared when the whole expression is squared?

The power of product rule states that when you raise a product to a power, each factor gets raised to that power. Think of it like distributing the exponent to everyone in the group!

Q: Does the order of variables matter in the final answer?

No! Since multiplication is commutative, $ c^2 \times b^2 \times a^2 $ equals $ a^2 \times b^2 \times c^2 $. Both arrangements are correct.

Q: What if there were more variables, like (abcde)²?

The same rule applies! Each variable gets the exponent: $ (abcde)^2 = a^2b^2c^2d^2e^2 $. The power of product rule works for any number of variables.

Q: How is this different from (a + b + c)²?

Completely different! With addition, you need to expand using FOIL or binomial methods. With multiplication, you simply apply the exponent to each factor separately.

Q: What if the exponent was 3 instead of 2?

Same process! $ (cba)^3 = c^3 \times b^3 \times a^3 $. The power of product rule works with any positive integer exponent.

Exponent Rules with Multiple Variables

Insert the corresponding expression:

$\left(c\times b\times a\right)^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 In order to open parentheses with a multiplication operation and an outside exponent

00:08 Raise each factor to the power

00:14 We will apply this formula to our exercise

00:25 This is one possible solution

00:26 In multiplication, the order of factors doesn't matter, therefore the expressions are equal

00:34 We can apply this formula to our exercise, and change the order of factors

00:42 This is another possible solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(c\times b\times a\right)^2=$

Step-by-step solution

Step 1: The problem provides the expression $\left(c \times b \times a\right)^2$ and asks us to expand it.
Step 2: We'll use the exponent rule for the power of a product, which states that $(xyz)^n = x^n \times y^n \times z^n$ . Applying this rule to our expression, we get:
$\left(c \times b \times a\right)^2 = c^2 \times b^2 \times a^2$
Since multiplication is commutative, the order of the factors doesn't affect the product. Therefore, the expression can also be written as:
$\left(c \times b \times a\right)^2 = a^2 \times b^2 \times c^2$

Therefore, the correct expressions are $c^2 \times b^2 \times a^2$ and $a^2 \times b^2 \times c^2$ .

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(abc)^n = a^n \times b^n \times c^n$
Technique: Apply exponent to each factor: $(c \times b \times a)^2 = c^2 \times b^2 \times a^2$
Check: Order doesn't matter due to commutative property: $c^2b^2a^2 = a^2b^2c^2$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one variable
Don't square just one variable like $c^2 \times b \times a$ = wrong result! This ignores the power of product rule and gives an incomplete answer. Always apply the exponent to every single variable in the product.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do all variables get squared when the whole expression is squared?

The power of product rule states that when you raise a product to a power, each factor gets raised to that power. Think of it like distributing the exponent to everyone in the group!

Does the order of variables matter in the final answer?

No! Since multiplication is commutative, $c^2 \times b^2 \times a^2$ equals $a^2 \times b^2 \times c^2$ . Both arrangements are correct.

What if there were more variables, like (abcde)²?

The same rule applies! Each variable gets the exponent: $(abcde)^2 = a^2b^2c^2d^2e^2$ . The power of product rule works for any number of variables.

How is this different from (a + b + c)²?

Completely different! With addition, you need to expand using FOIL or binomial methods. With multiplication, you simply apply the exponent to each factor separately.

What if the exponent was 3 instead of 2?

Same process! $(cba)^3 = c^3 \times b^3 \times a^3$ . The power of product rule works with any positive integer exponent.

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