Radicals

Definition

Radicals, in the context of simplifying and using square roots, refer to the square root symbol (√) and the expressions it represents. Radicals are used to denote the principal square root of a non-negative number, which is the positive value that, when multiplied by itself, equals the original number.

5 Must Know Facts For Your Next Test

1. Radicals can be simplified by finding the largest perfect square factor of the radicand and extracting it from the square root.
2. When the radicand is a fraction, the radical can be simplified by rationalizing the denominator, which involves moving the square root to the numerator.
3. Operations with radicals, such as addition, subtraction, multiplication, and division, can be performed by applying the properties of square roots.
4. Radicals can be used to represent the lengths of the sides of a right triangle, as in the Pythagorean Theorem.
5. Radicals are often used in various mathematical and scientific applications, including geometry, physics, and engineering.

Review Questions

• Explain the process of simplifying a radical expression by finding the largest perfect square factor of the radicand.
  • To simplify a radical expression, you first need to identify the largest perfect square factor of the radicand (the number inside the square root symbol). This can be done by finding the largest integer that, when multiplied by itself, equals the radicand. Once the largest perfect square factor is identified, it can be extracted from the square root, leaving a simplified radical expression. For example, to simplify $\sqrt{72}$, the largest perfect square factor of 72 is 4, so the expression can be rewritten as $2\sqrt{18}$, which is the simplified form.
• Describe the process of rationalizing the denominator of a radical fraction.
  • Rationalizing the denominator of a radical fraction involves moving the square root from the denominator to the numerator. This is done by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate is obtained by changing the sign of the square root term in the denominator. This process eliminates the square root in the denominator, resulting in a rational expression. For instance, to rationalize the fraction $\frac{1}{\sqrt{5}}$, you would multiply both the numerator and denominator by $\sqrt{5}$, yielding the simplified expression $\frac{\sqrt{5}}{5}$.
• Analyze how radicals are used to represent the lengths of the sides of a right triangle in the context of the Pythagorean Theorem.
  • The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This relationship can be expressed using radicals. Specifically, if the lengths of the two shorter sides of a right triangle are represented by the variables $a$ and $b$, then the length of the hypotenuse can be represented by the radical expression $\sqrt{a^2 + b^2}$. This connection between radicals and the Pythagorean Theorem is fundamental in various geometric and trigonometric applications, such as calculating the lengths of unknown sides in right triangles.

Related terms

Square Root: The square root of a number is the value that, when multiplied by itself, equals the original number. The square root symbol (√) is used to represent this operation.

Radicand: The radicand is the number or expression inside the square root symbol. It is the value whose square root is being calculated.

Perfect Square: A perfect square is a number that is the square of an integer, meaning it can be expressed as the product of two equal integers.