Mathematics » Roots and Radicals » Simplify and Use Square Roots

Simplifying and Using Square Roots Summary

Key Concepts

  • Simplified Square Root\(\sqrt{a}\) is considered simplified if \(a\) has no perfect-square factors.
  • Product Property of Square Roots If a, b are non-negative real numbers, then

    \(\sqrt{ab}=\sqrt{a}·\sqrt{b}\)

  • Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
    1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
    2. Use the product rule to rewrite the radical as the product of two radicals.
    3. Simplify the square root of the perfect square.
  • Quotient Property of Square Roots If a, b are non-negative real numbers and \(b\ne 0\), then

    \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)

  • Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
    1. Simplify the fraction in the radicand, if possible.
    2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
    3. Simplify the radicals in the numerator and the denominator.

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