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The distributive property is a fundamental concept in algebra that helps us simplify expressions by distributing terms across parentheses. In this exercise, it is applied to multiply a single term—\( \sqrt{5} \)—across two terms within the parentheses: \( \sqrt{10} \) and \( -\sqrt{x} \). By distributing \( \sqrt{5} \), we separate the expression as follows:

• \( \sqrt{5} \times \sqrt{10} \)
• \( \sqrt{5} \times -\sqrt{x} \)

This leads to a new expression: \( \sqrt{5} \times \sqrt{10} - \sqrt{5} \times \sqrt{x} \).
Remember, the distributive property works on the principle that multiplying a single term by each term inside the parentheses independently will simplify the equation. So, it is crucial to handle all terms without skipping any steps.

Surds, or irrational numbers that are expressed as roots, follow specific rules when multiplied. In this example, we use the rule \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This means if you multiply two roots together, you can combine what's inside the roots into one larger root.For instance:

• \( \sqrt{5} \times \sqrt{10} = \sqrt{50} \)
• \( \sqrt{5} \times \sqrt{x} = \sqrt{5x} \)

This simple rule helps us transition from a longer multiplication expression into a more manageable form. Just remember, when multiplying surds, multiply what's inside the square roots, not the roots themselves. Keeping track of these operations ensures accurate simplification as seen in this exercise.

Simplifying square roots involves breaking down the number inside the root to make it simpler and more readable. This often means finding factors that are perfect squares. In this specific example:

• The number 50 can be expressed as \( 25 \times 2 \).
• Since 25 is a perfect square, \( \sqrt{50} \) can further be written as \( \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} \).
• The square root of 25 is 5, so \( \sqrt{50} = 5\sqrt{2} \).

Using this method helps reduce complex expressions into simpler forms, which are easier to work with in further calculations or in evaluating problems. Identifying and factoring out perfect squares is key to mastering square root simplification. Practice will make this process second nature.