Complete the Expression: 8⁵ × b⁵ × z⁵ Product Problem

Q: Why can I combine these terms with the same exponent?

The Power of a Product Rule works in reverse! Since $ (abc)^n = a^n \times b^n \times c^n $, you can also go backwards: $ a^n \times b^n \times c^n = (abc)^n $.

Q: What if the exponents were different numbers?

If the exponents were different (like $ 8^3 \times b^5 \times z^2 $), you cannot use this rule! The power of a product rule only works when all exponents are the same.

Q: Can I write the answer without parentheses?

No! You need parentheses around $ (8 \times b \times z) $ to show that the entire product is raised to the 5th power. Without parentheses, only the last term would have the exponent.

Q: How do I know this is simpler than the original?

The condensed form $ (8bz)^5 $ is more compact and clearly shows the structure. It's like factoring - both forms are equal, but one is more organized.

Q: Does the order of multiplication inside the parentheses matter?

No! Because multiplication is commutative, $ (8 \times b \times z)^5 $, $ (b \times 8 \times z)^5 $, and $ (z \times b \times 8)^5 $ are all equivalent.

Power of a Product with Same Exponents

Insert the corresponding expression:

$8^5\times b^5\times z^5=$

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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 A multiplication where each factor is raised to the power of that factor (N)

00:07 Can be converted to parentheses of the entire multiplication raised to the power of the factor (N)

00:14 Apply this formula to our exercise

00:20 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:

$8^5\times b^5\times z^5=$

Step-by-step solution

To solve the problem, we need to condense the given expression using the rule for the power of a product.

Given is the expression: $8^5 \times b^5 \times z^5$ .

We notice that all the bases $8$ , $b$ , and $z$ have an exponent of 5.

By the Power of a Product Rule, which states $(a \times b \times c)^n = a^n \times b^n \times c^n$ , we can rewrite the product of these terms as a single product raised to the power of 5.

Therefore, the expression $8^5 \times b^5 \times z^5$ can be rewritten in a single exponent form as:

$\left(8 \times b \times z\right)^5$ .

The correct rewrite of the expression is $\left(8 \times b \times z\right)^5$ .

Final Answer

$\left(8\times b\times z\right)^5$

Key Points to Remember

Essential concepts to master this topic

Power of a Product Rule: When all terms have the same exponent, combine them
Technique: Rewrite $8^5 \times b^5 \times z^5$ as $(8 \times b \times z)^5$
Check: Expand back to verify: $(8bz)^5 = 8^5 \times b^5 \times z^5$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of using the power of a product rule
Don't think $8^5 \times b^5 \times z^5 = (8 \times b \times z)^{15}$ by adding 5+5+5! You're multiplying terms with the same exponent, not multiplying powers of the same base. Always use the power of a product rule: combine the bases and keep the same exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine these terms with the same exponent?

The Power of a Product Rule works in reverse! Since $(abc)^n = a^n \times b^n \times c^n$ , you can also go backwards: $a^n \times b^n \times c^n = (abc)^n$ .

What if the exponents were different numbers?

If the exponents were different (like $8^3 \times b^5 \times z^2$ ), you cannot use this rule! The power of a product rule only works when all exponents are the same.

Can I write the answer without parentheses?

No! You need parentheses around $(8 \times b \times z)$ to show that the entire product is raised to the 5th power. Without parentheses, only the last term would have the exponent.

How do I know this is simpler than the original?

The condensed form $(8bz)^5$ is more compact and clearly shows the structure. It's like factoring - both forms are equal, but one is more organized.

Does the order of multiplication inside the parentheses matter?

No! Because multiplication is commutative, $(8 \times b \times z)^5$ , $(b \times 8 \times z)^5$ , and $(z \times b \times 8)^5$ are all equivalent.

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