Perform the operation and/or simplify each of the following. Assume all variables represent positive real numbers. √50

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insert step 1> Start by identifying the perfect square factors of the number under the square root. In this case, 50 can be factored into 25 and 2, where 25 is a perfect square.

insert step 2> Rewrite the square root of 50 as the square root of the product of 25 and 2: \( \sqrt{50} = \sqrt{25 \times 2} \).

insert step 3> Use the property of square roots that states \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to separate the square root: \( \sqrt{50} = \sqrt{25} \times \sqrt{2} \).

insert step 4> Simplify \( \sqrt{25} \) since 25 is a perfect square: \( \sqrt{25} = 5 \).

insert step 5> Combine the simplified square root with the remaining factor: \( \sqrt{50} = 5 \times \sqrt{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, since 5 × 5 = 25. In algebra, square roots can be simplified by factoring the number into its prime factors and pairing them, which helps in simplifying expressions involving square roots.

Simplifying Radicals

Simplifying radicals involves rewriting a square root in its simplest form. This is done by identifying perfect squares within the radicand (the number under the square root) and extracting them. For instance, √50 can be simplified to √(25 × 2), which equals 5√2, as 25 is a perfect square.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with square roots, as they can be expressed in exponential form. The square root of a number can be represented as that number raised to the power of 1/2. This property aids in manipulating and simplifying expressions involving roots and powers effectively.