Outro W12

8.9 Use Direct and Inverse Variation

9.1 Simplify and Use Square Roots

• Note that the square root of a negative number is not a real number.
• Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.
• We can estimate square roots using nearby perfect squares.
• We can approximate square roots using a calculator.
• When we use the radical sign to take the square root of a variable expression, we should specify that x≥0$x\ge 0$ to make sure we get the principal square root.

9.2 Simplify Square Roots

• Simplified Square Root $\sqrt{a}$ is considered simplified if a$a$ 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. Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
  2. Step 2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Step 3. Simplify the square root of the perfect square.
• Quotient Property of Square Roots If a, b are non-negative real numbers and b≠0$b\ne 0$, 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. Step 1. Simplify the fraction in the radicand, if possible.
  2. Step 2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
  3. Step 3. Simplify the radicals in the numerator and the denominator.

9.3 Add and Subtract Square Roots

• To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
• Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.