4-2 Nth Roots

1. 4-2 Nth Roots Radical and Rational Forms Kyle McFarlane

2. Radical Form • The radical form is more commonly seen in mathematics. It is made up of an index, radical sign, and a radicand. 3√x4 Exponent Index Radical Sign Radicand

3. Rational Form • The rational form is less seen in everyday mathematics. It is made up of a base, index and exponent. x 4 Exponent 3 Index Base Note: 3√x4 = x4/3

4. Examples: Radical to Rational • Describe√16 in its rational form. • Step 1- The radicand (16) becomes the base. • Step 2- The index becomes the denominator and the exponent becomes the numerator. In this case since the index and the exponent are missing we can assume the index is 2 and the exponent is 1. • Step 3- Write the rational form. • Answer: 161/2

5. Examples: Radical to Rational • Describe 3√3x2 in its rational form. • Step 1- The radicand (3x) becomes the base. Any numbers before the variable (x) must be included in the base. • Step 2- The index (3) becomes the denominator and the exponent (2) becomes the numerator. • Step 3- Write the rational form. • Answer: 3x2/3

6. Examples: Radical to Rational • Describe √(64x36y128)in its rational form. • Step 1- Since there are multiple variables in the radicand we must treat each of them separately. • Step 2 - The index is missing so we must assume 2. The square root of 64 = 8. • Step 3- Now rationalize x. 2√x36 becomes x36/2 . This can further be simplified to x18 . • Step 4- Now rationalize y. 2√y128 becomes y128/2 . Which can further simplified to y64 . • Answer: 8x18y64 Note: You can simply divide the exponent of x and y by the index as a shortcut to the answer. Don’t forget that numbers must be resolved too e.g. the square root of 64 is 8.

7. Examples: Rational to Radical • Describe271/3 in its radical form. • Step 1- The base (27) becomes the radicand. • Step 2-The numerator (1) becomes the exponent, and the denominator (3) becomes the index. • Step 3- Write the radical form. • Answer: 3√271 or simply 3√27. • To solve: 27=333. Since we have three 3’s they can be removed from the radical so the answer is 3.

8. Examples: Rational to Radical • Describe4x5/12 in its radical form. • Step 1- The base (4x) becomes the radicand. • Step 2-The numerator (5) becomes the exponent, and the denominator (12) becomes the index. • Step 3- Write the radical form. • Answer: 12√4x5 .

9. Examples: Rational to Radical • Describe4x3/4y5/4 in its radical form. • Step 1- Since there are multiple variables we must treat each one separately. • Step 2-Now radicalize x. The numerator 3 becomes the exponent and the denominator becomes the index. 4√x3 • Step 3- Now radicalize y. The numerator becomes the exponent and the denominator becomes the index. 4√y5 • Step 4- Since the index of x and y is the same (4) we can combine them and also include the 4 but we must raise the 4 to the power of 4. • Answer: 4√(256x3y5)