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Simplifying Radicals: Square Root of 75 and Square Root of 27

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Simplifying Radicals: Square Root of 75 and Square Root of 27

In the realm of mathematics, simplifying radicals is a fundamental skill that allows us to express numbers in their most concise form. Radicals, often referred to as roots, represent the inverse operation of exponentiation. For instance, the square root of a number is the value that, when multiplied by itself, equals the original number.

Let’s delve into the simplification of two common radicals: the square root of 75 and the square root of 27. We’ll break down the process step-by-step to make it clear and understandable.

Simplifying the Square Root of 75

  1. Find the largest perfect square factor of 75. A perfect square is a number that can be obtained by squaring an integer. In this case, the largest perfect square factor of 75 is 25 (5 x 5 = 25).
  2. Rewrite 75 as the product of the perfect square and another factor. 75 can be expressed as 25 x 3.
  3. Apply the square root property: √(a x b) = √a x √b. Therefore, √75 = √(25 x 3) = √25 x √3.
  4. Simplify. √25 = 5, so √75 = 5√3.

Simplifying the Square Root of 27

  1. Find the largest perfect square factor of 27. The largest perfect square factor of 27 is 9 (3 x 3 = 9).
  2. Rewrite 27 as the product of the perfect square and another factor. 27 can be expressed as 9 x 3.
  3. Apply the square root property: √(a x b) = √a x √b. Therefore, √27 = √(9 x 3) = √9 x √3.
  4. Simplify. √9 = 3, so √27 = 3√3.

Understanding the Concept

Simplifying radicals is essentially about breaking down a radical into its simplest form. We achieve this by extracting the largest perfect square factor from the radicand (the number under the radical sign). By doing so, we can express the radical as a product of a whole number and a simplified radical.

Key Points to Remember

  • Always look for the largest perfect square factor to ensure the most simplified form.
  • The square root property (√(a x b) = √a x √b) is crucial for simplifying radicals.
  • Practice is key to mastering the simplification of radicals.

By understanding the concept and applying the steps outlined above, you can confidently simplify radicals like the square root of 75 and the square root of 27. This skill is essential for various mathematical operations and problem-solving in algebra and beyond.

Simplification