Perform the indicated operations. Assume all variables represent positive real numbers. 8√(2x) - √(8x) + √(72x)

Verified step by step guidance

1. Rewrite each radical expression to simplify the square roots by factoring out perfect squares. For example, express each radicand as a product of a perfect square and another factor: \$8\sqrt{2x}\(, \)\sqrt{8x}\(, and \)\sqrt{72x}$.
2. Simplify each square root separately by taking the square root of the perfect square factor out of the radical. For instance, \(\sqrt{8x} = \sqrt{4 \cdot 2x} = 2\sqrt{2x}\).
3. After simplifying, rewrite the entire expression with the simplified radicals. This will give you terms all involving \(\sqrt{2x}\), making it easier to combine like terms.
4. Combine like terms by adding or subtracting the coefficients of \(\sqrt{2x}\). Remember to keep the radical part unchanged while combining the coefficients.
5. Write the final simplified expression as a single term involving \(\sqrt{2x}\) with the combined coefficient.

Key Concepts

Simplifying Square Roots

Simplifying square roots involves expressing the radicand as a product of perfect squares and other factors. For example, √72x can be broken down into √(36*2*x) = 6√(2x). This process helps in combining like terms and making expressions easier to work with.

Like Terms with Radicals

Like terms in radical expressions have the same radicand. For instance, terms containing √(2x) can be combined by adding or subtracting their coefficients. Recognizing and grouping like radical terms is essential for simplifying expressions.

Operations with Radicals

Performing addition and subtraction with radicals requires first simplifying each radical and then combining like terms. Since variables represent positive real numbers, we can treat the radicals as positive quantities, ensuring valid operations without considering absolute values.