Write the following expression in simplified radical form: [tex]$\sqrt[3]{40 x^{10} z^{18}}$[/tex]
Answer (2)
Factor the constant: Rewrite 40 as 2 3 c d o t 5 .
Separate the terms: Apply the property 3 ab = 3 a ⋅ 3 b .
Simplify each cube root: Simplify 3 2 3 , 3 x 10 , and 3 z 18 .
Combine the terms: Write the final simplified expression as 2 x 3 z 6 3 5 x .
Explanation
Understanding the Problem We are asked to simplify the radical expression 3 40 x 10 z 18 . This involves finding the largest perfect cube factors of the terms inside the cube root and taking them out. Let's break down the expression step by step.
Factoring the Constant First, we factor the constant 40. The prime factorization of 40 is 2 3 ⋅ 5 = 8 ⋅ 5 . Thus, we can rewrite the expression as 3 2 3 ⋅ 5 ⋅ x 10 ⋅ z 18 .
Separating the Terms Next, we use the property n ab = n a ⋅ n b to separate the terms: 3 2 3 ⋅ 5 ⋅ x 10 ⋅ z 18 = 3 2 3 ⋅ 3 5 ⋅ 3 x 10 ⋅ 3 z 18 .
Simplifying Cube Roots Now, we simplify each cube root individually. We have:\begin{itemize} \item 3 2 3 = 2 \item 3 5 remains as 3 5 since 5 has no perfect cube factors. \item For 3 x 10 , we can rewrite x 10 as x 9 ⋅ x , where x 9 = ( x 3 ) 3 is a perfect cube. Thus, 3 x 10 = 3 x 9 ⋅ x = 3 x 9 ⋅ 3 x = x 3 3 x .
\item For 3 z 18 , we note that z 18 = ( z 6 ) 3 , so 3 z 18 = z 6 .
\end{itemize}
Combining the Terms Substituting these simplified terms back into the expression, we get: 2 ⋅ 3 5 ⋅ x 3 3 x ⋅ z 6 = 2 x 3 z 6 3 5 x .
Final Answer Therefore, the simplified radical expression is 2 x 3 z 6 3 5 x .
Examples
Radical expressions are used in various fields, such as physics and engineering, to simplify complex formulas and calculations. For example, when calculating the period of a pendulum, the formula involves a square root. Simplifying such expressions makes it easier to understand and apply the formula in practical situations, such as designing clocks or analyzing oscillatory motion. Similarly, in signal processing, simplifying radical expressions can help in analyzing and manipulating signals more efficiently.
To simplify 3 40 x 10 z 18 , we factor 40 as 2 3 ⋅ 5 and separately simplify the cube roots of each term. The final simplified expression is 2 x 3 z 6 3 5 x .
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