Question

Simplify the radical expression. $$\sqrt{162}$$

Short answer

\(\sqrt{162}\) simplifies to \(9\sqrt{2}\).

Step-by-step solution

Prime Factorization of 162

Start by performing a prime factorization of the number 162. The prime factorization of 162 is \(2 \times 3 \times 3 \times 3 \times 3\), or more conveniently expressed as \(2 \times (3^4)\). This helps to identify the perfect squares which can be simplifed under the square root.

Apply Square Root Properties

Next, apply square root properties to simplify the expression. The square root of a product can be expressed as the product of the square roots of the factors. So, \(\sqrt{162}\) can be rewritten as \(\sqrt{2 \times (3^4)}\), which further simplifies to \(\sqrt{2} \times \sqrt{(3^4)}\). Calculating square root of \(3^4\) gives 9 (since \(9^2 = 3^4\)). So, the simplified form would be \(9\sqrt{2}\).

Final Simplified Form

Therefore, the final simplified form of \(\sqrt{162}\) is \(9\sqrt{2}\). This simplification process helps to reduce the complexity of the radical, making it easier to deal with in subsequent computations, if needed.