Expand the Expression: Finding (b×z×a)^5

Q: Why does the exponent apply to all three variables?

The power of a product rule states that $ (xyz)^n = x^n \times y^n \times z^n $. Since all variables are multiplied together inside parentheses, the exponent 5 must be applied to each one.

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

No! Since multiplication is commutative, you can write the variables in any order. $ a^5b^5z^5 $, $ b^5z^5a^5 $, and $ z^5a^5b^5 $ are all correct. Alphabetical order is just a common convention.

Q: What if I had different exponents for each variable originally?

If you started with something like $ (a^2 \times b^3)^5 $, you would multiply the exponents: $ a^{2×5} \times b^{3×5} = a^{10} \times b^{15} $. But in this problem, each variable has an implied exponent of 1.

Q: How can I remember this rule?

Think of it as distributing the outside exponent to everyone inside! Just like distributing cookies to each person in a group - the exponent 5 gets "given" to each variable: a, b, and z.

Q: What's the difference between this and adding exponents?

You add exponents when multiplying the same base: $ a^2 \times a^3 = a^5 $. You apply exponents to each factor when raising a product to a power. These are completely different situations!

Power of Products with Multiple Variables

Insert the corresponding expression:

$\left(b\times z\times a\right)^5=$

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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 with a multiplication operation and an outside exponent

00:07 We raise each factor to the power

00:13 We will apply this formula to our exercise

00:24 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:

$\left(b\times z\times a\right)^5=$

Step-by-step solution

To solve the problem, we will use the rule of exponents known as the power of a product rule, which states that for any real numbers or expressions $x$ , $y$ raised to a power $n$ , the following holds:

$(x \times y)^n = x^n \times y^n$ .

We have the expression $\left(b \times z \times a\right)^5$ . According to the power of a product rule, we apply the exponent 5 to each factor inside the parenthesis.

Let's break it down:

Apply $5$ to $b$ : $(b)^5 = b^5$ .
Apply $5$ to $z$ : $(z)^5 = z^5$ .
Apply $5$ to $a$ : $(a)^5 = a^5$ .

By applying the exponent to each factor, we obtain:
$(b \times z \times a)^5 = b^5 \times z^5 \times a^5$ .

Since multiplication is commutative, we can write it in any order, and a common convention is ordering it alphabetically:

Thus, $a^5 \times b^5 \times z^5$ is the simplified expression.

Therefore, the correct answer to the problem is $a^5 \times b^5 \times z^5$ , which corresponds to choice 1.

Final Answer

$a^5\times b^5\times z^5$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, apply the exponent to each factor
Technique: $(b \times z \times a)^5 = b^5 \times z^5 \times a^5$
Check: Verify each variable has the same exponent as the original power ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to one factor
Don't raise just one variable to the 5th power like $a^5 \times b \times z$ = wrong answer! This ignores the power of a product rule and gives an incomplete result. Always apply the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

Why does the exponent apply to all three variables?

The power of a product rule states that $(xyz)^n = x^n \times y^n \times z^n$ . Since all variables are multiplied together inside parentheses, the exponent 5 must be applied to each one.

Does the order of the variables matter in my final answer?

No! Since multiplication is commutative, you can write the variables in any order. $a^5b^5z^5$ , $b^5z^5a^5$ , and $z^5a^5b^5$ are all correct. Alphabetical order is just a common convention.

What if I had different exponents for each variable originally?

If you started with something like $(a^2 \times b^3)^5$ , you would multiply the exponents: $a^{2×5} \times b^{3×5} = a^{10} \times b^{15}$ . But in this problem, each variable has an implied exponent of 1.

How can I remember this rule?

Think of it as distributing the outside exponent to everyone inside! Just like distributing cookies to each person in a group - the exponent 5 gets "given" to each variable: a, b, and z.

What's the difference between this and adding exponents?

You add exponents when multiplying the same base: $a^2 \times a^3 = a^5$ . You apply exponents to each factor when raising a product to a power. These are completely different situations!

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