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Radical expressions involve roots, such as square roots or cube roots. The expression \( \sqrt{50x} - \sqrt{8x} \) involves square roots, which are a specific kind of radical expression. A radical expression consists of a root symbol, often with a number underneath it called the radicand. Here, \( 50x \) and \( 8x \) are the radicands. Radical expressions can often be simplified by manipulating the radicands. This involves breaking down the radicands into their prime factors, to see if any portions can be taken out from under the radical. For example, \( \sqrt{50x} \) can be simplified by recognizing that 50 is \( 2 \times 5^2 \).

• Take out pairs of any numbers: In \( 50 \), since \( 5^2 \) is a perfect square, \( 5 \) can be removed from under the square root.
• Simplify to \( 5\sqrt{2x} \): Taking out the \( 5 \) leaves \( 2x \) under the radical.

Doing similar operations on \( \sqrt{8x} \) results in \( 2\sqrt{2x} \). This simplification process is crucial for combining and simplifying radical expressions later, as it often turns a seemingly complex problem into a more manageable one.

Like terms are terms that contain the same variable raised to the same power. When dealing with radical expressions, like terms will have the same index and radicand. In our given problem, \( 5\sqrt{2x} \) and \( 2\sqrt{2x} \) are like terms as they both contain the radical \( \sqrt{2x} \).Understanding like terms is essential because it allows us to combine radical expressions under certain conditions. For example, just like combining \( 2x \) and \( 3x \) to get \( 5x \), you can combine \( 5\sqrt{2x} \) and \(-2\sqrt{2x} \) by adding or subtracting the coefficients.

• Ensure radicals have the same radicand and root index.
• Combine the coefficients: here it would be \( 5 - 2 = 3 \).
• The result is \( 3\sqrt{2x} \), showing that simplification resulted in the same radicand within combined terms.

This process can greatly simplify expressions and enable easier interpretations or further calculations.

Prime factorization is a method of simplifying numbers into their prime number constituents. This is integral in simplifying radicals, especially when seeking to turn radicands into simpler terms.Consider the number 50, which is broken down into primes as \( 50 = 2 \times 5 \times 5 \) or \( 2 \times 5^2 \). Similarly, \( 8 = 2^3 \). The method of prime factorization helps identify repeated factors that can be removed from under the radical sign.

• Identify repeated factors: In \( 50x \), the \( 5^2 \) allows us to take a \( 5 \) out of the radical.
• Leaves \( 2x \) as the new radicand, simplifying the radical to \( 5\sqrt{2x} \).
• For \( 8x \), \( 2^3 \) simplifies the radical to \( 2\sqrt{2x} \).

By using prime factorization, you make the radicals easier to work with, as illustrated in our original exercise. It is a key skill in both simplifying and recognizing like terms in radical expressions.