Questions: Simplify. ∛(27 y^18) Assume that the variable represents a positive real number.

Transcript text: Simplify. \[ \sqrt[3]{27 y^{18}} \] Assume that the variable represents a positive real number.

Solution

Solution Steps

To simplify the expression \(\sqrt[3]{27 y^{18}}\), we need to find the cube root of each component separately. The cube root of 27 is 3, and the cube root of \(y^{18}\) is \(y^{18/3} = y^6\). Therefore, the simplified expression is \(3y^6\).

Step 1: Find the Cube Root of 27

To simplify the expression \(\sqrt[3]{27 y^{18}}\), we first calculate the cube root of 27. Since \(27 = 3^3\), we have: \[ \sqrt[3]{27} = 3 \]

Step 2: Find the Cube Root of \(y^{18}\)

Next, we find the cube root of \(y^{18}\). Using the property of exponents, we can express this as: \[ \sqrt[3]{y^{18}} = y^{\frac{18}{3}} = y^6 \]

Step 3: Combine the Results

Now, we combine the results from Steps 1 and 2: \[ \sqrt[3]{27 y^{18}} = \sqrt[3]{27} \cdot \sqrt[3]{y^{18}} = 3 \cdot y^6 \]

Final Answer

Thus, the simplified expression is: \[ \boxed{3y^6} \]

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