Complete the Expression: (z×b)⁴ Fourth Power Problem

Q: Do I multiply the exponents or keep them the same?

You keep the exponent the same for each factor. The 4 gets distributed to both z and b, so you get $ z^4 \times b^4 $, not $ z^8 \times b^8 $.

Q: What's the difference between (z×b)⁴ and z×b⁴?

Huge difference! $ (z \times b)^4 = z^4 \times b^4 $ (both variables to the 4th), but $ z \times b^4 $ means only b is raised to the 4th power.

Q: Can I write the answer as (zb)⁴ instead?

While $ (zb)^4 $ means the same thing as $ (z \times b)^4 $, the expanded form $ z^4 \times b^4 $ is what's typically expected as the final answer.

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

Yes! For example, $ (xyz)^4 = x^4 \times y^4 \times z^4 $. The power distributes to every single factor inside the parentheses.

Power of Products with Fourth Exponents

Insert the corresponding expression:

$\left(z\times b\right)^4=$

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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 Raise each factor to the power

00:13 We will apply this formula to our exercise

00:19 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:

$\left(z\times b\right)^4=$

Step-by-step solution

To solve the expression $\left(z \times b\right)^4$ , we will apply the Power of a Product rule, which states that $(ab)^n = a^n \times b^n$ .

Step 1: Identify the base and the exponent in the expression. Here, the base is $z \times b$ and the exponent is 4.
Step 2: Apply the Power of a Product rule to distribute the exponent to each factor inside the parentheses.
Step 3: Raise each variable to the power of 4:
- $z^4$
- $b^4$
Step 4: Multiply the results together:
- $z^4 \times b^4$

Therefore, the expanded form of $\left(z \times b\right)^4$ is $z^4 \times b^4$ .

Final Solution: $z^4 \times b^4$

Final Answer

$z^4\times b^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to each factor: $(ab)^n = a^n \times b^n$
Technique: Distribute the 4: $(z \times b)^4 = z^4 \times b^4$
Check: Verify each variable has the same exponent as original: both $z^4$ and $b^4$ ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to one factor
Don't write $(z \times b)^4 = z^4 \times b$ or $z \times b^4$ ! This violates the power rule and gives an incorrect expression. Always distribute 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 can't I just multiply z×b×4 instead of raising to the fourth power?

Exponents mean repeated multiplication, not regular multiplication! $(z \times b)^4$ means (z×b) multiplied by itself 4 times, not z×b×4.

Do I multiply the exponents or keep them the same?

You keep the exponent the same for each factor. The 4 gets distributed to both z and b, so you get $z^4 \times b^4$ , not $z^8 \times b^8$ .

What's the difference between (z×b)⁴ and z×b⁴?

Huge difference! $(z \times b)^4 = z^4 \times b^4$ (both variables to the 4th), but $z \times b^4$ means only b is raised to the 4th power.

Can I write the answer as (zb)⁴ instead?

While $(zb)^4$ means the same thing as $(z \times b)^4$ , the expanded form $z^4 \times b^4$ is what's typically expected as the final answer.

Does this rule work with more than two variables?

Yes! For example, $(xyz)^4 = x^4 \times y^4 \times z^4$ . The power distributes to every single factor inside the parentheses.

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