Questions: Simplify the radical expression (15 - √162) / 3. Write your answer in exact form without decimals.

Solution Steps

To simplify the given radical expression, we need to break down the square root term and simplify the fraction. First, simplify the square root of 162 by expressing it in terms of its prime factors. Then, simplify the entire expression by dividing each term in the numerator by the denominator.

First, we simplify the square root term \(\sqrt{162}\). We can express 162 as a product of its prime factors: \[ 162 = 2 \times 81 = 2 \times 3^4 \] Thus, \[ \sqrt{162} = \sqrt{2 \times 3^4} = 3^2 \times \sqrt{2} = 9\sqrt{2} \]

Next, we substitute \(\sqrt{162}\) with \(9\sqrt{2}\) in the original expression: \[ \frac{15 - \sqrt{162}}{3} = \frac{15 - 9\sqrt{2}}{3} \]

We divide each term in the numerator by the denominator: \[ \frac{15}{3} - \frac{9\sqrt{2}}{3} = 5 - 3\sqrt{2} \]

Final Answer