Pre Ap Algebra 2 Lesson 1 2 Pair Work Answer Key - Fill and Sign ... - Free Printable

It looks like you've uploaded a document related to Pre-Algebra 2, Unit 1 – Rational Exponents, and it contains a homework assignment with a problem involving rational exponents. Let's go through the problem step by step.

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🔍 Problem Summary:

You are given an expression:
\[
(32)^{\frac{1}{5}} \cdot (64)^{\frac{1}{6}}
\]

And asked:
What is the value of this expression?

Additionally, there’s a section asking:
> "What about these complex fractions?"
With options:
- a) \( \frac{1}{a^3} \)
- b) \( \frac{1}{a^4} \)
- c) \( \frac{1}{a^5} \)
- d) \( \frac{1}{a^6} \)

But the context suggests that this might be part of a larger set of problems — let’s focus on the main one first.

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## ✔ Step-by-Step Solution

We are evaluating:
\[
(32)^{\frac{1}{5}} \cdot (64)^{\frac{1}{6}}
\]

Step 1: Express 32 and 64 as powers of 2

- \( 32 = 2^5 \)
- \( 64 = 2^6 \)

So substitute:
\[
(2^5)^{\frac{1}{5}} \cdot (2^6)^{\frac{1}{6}}
\]

Step 2: Use exponent rule: \( (a^m)^n = a^{m \cdot n} \)

Apply the power of a power rule:

- \( (2^5)^{\frac{1}{5}} = 2^{5 \cdot \frac{1}{5}} = 2^1 = 2 \)
- \( (2^6)^{\frac{1}{6}} = 2^{6 \cdot \frac{1}{6}} = 2^1 = 2 \)

Step 3: Multiply the results

\[
2 \cdot 2 = 4
\]

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✔ Final Answer:

\[
\boxed{4}
\]

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## 🧩 Bonus: Understanding the Complex Fractions

The second part asks about “complex fractions” like:
- \( \frac{1}{a^3}, \frac{1}{a^4}, \ldots \)

This likely refers to simplifying expressions such as:
\[
\left( \frac{1}{a} \right)^3 = \frac{1}{a^3}, \quad \left( \frac{1}{a} \right)^4 = \frac{1}{a^4}, \text{ etc.}
\]

Or perhaps simplifying expressions like:
\[
a^{-n} = \frac{1}{a^n}
\]

So if you were asked to simplify something like \( a^{-3} \), the answer would be \( \frac{1}{a^3} \), which matches option (a).

But since no specific expression was given for this part, we assume it's just reinforcing the idea that negative exponents become reciprocals.

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## ✔ Conclusion

The main problem:
\[
(32)^{\frac{1}{5}} \cdot (64)^{\frac{1}{6}} = 2 \cdot 2 = \boxed{4}
\]

Let me know if you'd like help with any other parts of the worksheet or clarification on rational exponents!