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

$112^0=\text{?}$

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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