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Chapter 2: Problem 5

Simplify. \(\left(5 z^{4}\right)^{2}\left(2 z^{6}\right)^{3}\)

Short Answer

Expert verified
The simplified expression is \(200z^{26}\).

Step by step solution

01

Simplify the exponential terms separately

First, we will simplify \(\left(5 z^{4}\right)^{2}\) and \(\left(2 z^{6}\right)^{3}\) using the exponent rule: \((ab)^n = a^n b^n\) For \(\left(5 z^{4}\right)^{2}\): \[(5 z^{4})^{2} = 5^2 (z^{4})^{2}\] For \(\left(2 z^{6}\right)^{3}\): \[(2 z^{6})^{3} = 2^3 (z^{6})^{3}\]
02

Evaluate the exponential terms

Now, we will evaluate the exponential terms found in the previous step. For \(\left(5 z^{4}\right)^{2}\): \[5^2 (z^{4})^{2} = 25z^{8}\] For \(\left(2 z^{6}\right)^{3}\): \[2^3 (z^{6})^{3} = 8z^{18}\]
03

Multiply the simplified terms

Now, we will multiply the simplified terms together: \[25z^{8} \cdot 8z^{18}\] Use the rule for multiplication of exponents with the same base: \(a^{m} \cdot a^{n} = a^{m+n}\) \[25z^{8} \cdot 8z^{18} = 25 \cdot 8 z^{8+18} = 200z^{26}\] So, the simplified expression is: \[200z^{26}\]

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