Simplifying Radicals Section 10-2 Part 2

Goals Goal Rubric To simplify radicals involving quotients. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Rationalize the Denominator

Simplest Form

Simplifying Radicals Simplifying Fractions within radicals The radicand has no fractions. Simplifying Radicals Simplifying Fractions within radicals

Dividing Radicals

Example: Simplifying Fractions Within Radicals B. Simplify. Quotient Property of Square Roots. Quotient Property of Square Roots. Simplify. Simplify.

Your Turn: Simplify. a. b. Quotient Property of Square Roots.

Your Turn: Simplify. Quotient Property of Square Roots. Factor the radicand using perfect squares. Simplify.

Example: Simplifying Radicals with Quotients Quotient Property. Product Property. Write 108 as 36(3). Simplify.

Example: Simplifying Radicals with Quotients Quotient Property. Product Property. Simplify.

Your Turn: Simplify. Quotient Property. Product Property. Write 20 as 4(5). Simplify.

Your Turn: Simplify. Quotient Property. Product Property. Write as .

Your Turn: Simplify. Quotient Property. Simplify.

Simplifying Radicals Rationalizing Denominators No radicals appear in the denominator of a fraction. Simplifying Radicals Rationalizing Denominators

Definition Rationalize the Denominator – the process of removing a radical from the denominator of a fraction. A quotient with a square root in the denominator is not simplified. To simplify these expressions, multiply by a form of 1 to get a perfect-square radicand in the denominator. This is called rationalizing the denominator. Why - It is easier to work with radical expressions if the denominators do not contain any radicals.

Rationalizing the Denominator The process of multiplying the top and bottom of a radical expression by a form of 1to make the denominator a perfect square is called rationalizing the denominator. Any number divided by itself is 1. Multiplication by 1 does not change the value of the expression. The denominator contains no radicals, we have rationalized the denominator.

Example: Rationalize Denominator Notice what we multiplied by to get rid of the square root of 2 in the denominator. When we multiply a square root by itself we get an answer containing no radicals. For monomial denominators containing square roots, your choice for multiplication can just be that square root over itself. But you should simplify first as we did here – or your problem may be messier than it needs to be. The denominator contains no radicals, we have rationalized the denominator.

Example: Rationalizing the Denominator Factors of 56. Simplify denominator. Multiply numerator and denominator by the denominator Take square root of 4 and write it outside the radical sign Simplify Simplify…divide 2 and 8 3 examples – key is you can’t leave a radical in the denominator. Introduce by asking what number can you multiply anything by and its identity doesn’t change. (1)

Example: Rationalizing the Denominator Simplify denominator. Break apart 504. Take square root of y10 and write it outside the radical sign Simplify Multiply numerator and denominator by the denominator Take square root of 36 3 examples – key is you can’t leave a radical in the denominator. Simplify Simplify Simplify

Your Turn: Simplify the expression.

Your Turn: Simplify the expression. A. B. C.

Your Turn: Simplify the expression. Simplify denominator. Simplify the numerator. Multiply numerator and denominator by the denominator 3 examples – key is you can’t leave a radical in the denominator. Simplify the terms outside the radical sign. Simplify

Your Turn: Simplify the expression. Simplify denominator. Simplify the numerator. Multiply numerator and denominator by the denominator 3 examples – key is you can’t leave a radical in the denominator. Simplify the terms outside the radical sign. Simplify

Review: Simplifying Radicals Steps Requirements No perfect square factors other than 1 are in the radicand. No fractions are in the radicand. No radicals appear in the denominator of a fraction. Reduce the fraction, if possible. Split the radical. Square root perfect squares. Look for perfect square factors and pull them out. Rationalize the denominator, if needed. Simplify if possible.

Assignment 10-2 Pt 2 Exercises Pg. 616 - 617: #8 – 26 even