11-1: Simplifying Radicals Essential question: How do you simplify a radical?

11-1: Simplifying Radicals Radical expressions like contain a radical. You read as “the square root of the quantity x plus 3” You can simplify radicals by removing perfect-square factors from underneath the radical (called the “radicand”). Multiplication Property of Square Roots For every number a > 0 and b > 0, Example:

11-1: Simplifying Radicals Example 1: Simplify Method A (Finding Perfect Squares) Perfect squares are numbers that can be produced by multiplying a number by itself (e.g. 25 is a perfect square since 25 = 5 ● 5) Find any perfect squares that can divide the radicand, and rewrite the radicand as the product of two radicals Since 192 = 64 ● 3,

11-1: Simplifying Radicals Example 1: Simplify Method B (Using Factor Trees) Break down the number under the radical into its prime factorization (like we did with finding a GCF) If there are any pairs, they merge together for one to come outside the radical Numbers brought outside the radical get multiplied together, as well as numbers left inside the radical

11-1: Simplifying Radicals Your Turn Simplify each radical

11-1: Simplifying Radicals You can simplify radical expressions that contain variables. A variable with an even exponent is a perfect square (e.g. x8 = x4 ● x4) A variable with an odd exponent (other than 1 or -1) is the product of a perfect square and the variable (e.g. x7 = x3 ● x3 ● x) You can also break variable down (like we did with GCFs) and remove pairs as a singular entity In Algebra 1, we will assume that all variables under radicands represent positive numbers

11-1: Simplifying Radicals Example 2: Simplify 45 = 3 ● 3 ● 5 a5 = a2 ● a2 ● a YOUR TURN Simplify each radical

11-1: Simplifying Radicals You can use the multiplication property to write Often times, it’s easier to multiply two radicals together and then simplify. Example 3: Multiplying Two Radicals

11-1: Simplifying Radicals Your Turn Simplify each radical

11-1: Simplifying Radicals Example 4: Real-World Application You can use the formula to estimate the distance d in miles to a horizon when h is the height of the viewer’s eyes above the ground in feet. Suppose you are looking out on a second floor window 25 feet above the ground. Find the distance you can see to the horizon. Round your answer to the nearest mile. You can see about 6 miles.

11-1: Simplifying Radicals Your Turn Suppose you are looking out a fourth floor window 52 ft above the ground. Use the formula to estimate the distance you can see to the horizon. Round your answer to the nearest mile. 9 miles

11-1: Simplifying Radicals Division Property of Square Roots For every number a > 0 and b > 0, Example:

11-1: Simplifying Radicals When the denominator is a perfect square, it’s easier to simplify separately Example 5: Simplify each expression A) B)

11-1: Simplifying Radicals Your Turn Simplify each expression A) B) C)

11-1: Simplifying Radicals When the denominator is not a perfect square, it may be easier to divide first and then simplify Example 6: Simplify each expression A) B)

11-1: Simplifying Radicals Your Turn Simplify each expression A) B) C)

11-1: Simplifying Radicals Sometimes, the denominator may not come out to be a perfect square. Fractions are in their simplest form when there are no roots in the denominator, which means you may have to rationalize a denominator. To rationalize, you multiply the numerator AND denominator by the whatever radical expression is on the denominator of the fraction.

11-1: Simplifying Radicals Example 7a: Rationalizing a Denominator Multiply by to make a perfect square Multiply square roots Simplify

11-1: Simplifying Radicals Example 7b: Rationalizing a Denominator Simplify the denominator first Multiply by Multiply numerator & denominator Simplify

11-1: Simplifying Radicals Your Turn Simplify by rationalizing the denominator A) B) C)

11-1: Simplifying Radicals Assignment Worksheet #11-1 1 – 81 (odds) Due 5/26 (next Tuesday)