Simplify each radical. Assume all variables represent positive real numbers. √192

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Start by expressing the number inside the radical, 192, as a product of its prime factors. This helps identify perfect squares. For 192, find its prime factorization.

Rewrite the square root using the prime factorization: $\sqrt{192} = \sqrt{2^6 \times 3}$, since 192 can be factored into \$2^6$ and $3$.

Apply the property of square roots that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. So, separate the radical into $\sqrt{2^6} \times \sqrt{3}$.

Simplify $\sqrt{2^6}$ by recognizing that $\sqrt{2^6} = 2^{6/2} = 2^3$, because the square root of a power is the power divided by 2.

Combine the simplified parts to write the expression as $2^3 \times \sqrt{3}$, or equivalently $8\sqrt{3}$, which is the simplified form of $\sqrt{192}$.

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Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring the radicand into perfect squares and other factors. For example, √192 can be broken down into √(64 × 3), where 64 is a perfect square, allowing simplification to 8√3.

Perfect Squares

Perfect squares are numbers that are squares of integers, such as 1, 4, 9, 16, 25, and so on. Recognizing perfect squares within a radicand helps simplify radicals by extracting their square roots as whole numbers.

Properties of Square Roots

The property √(a × b) = √a × √b allows the separation of a square root of a product into the product of square roots. This property is essential for breaking down complex radicals into simpler parts for easier simplification.