√To Simplify (Or Unsimplify) Different Roots√

Simplifying Square Roots:

Step 1: List out factors of the radicand (number under the square root. Step 2: Determine which factors are perfect squares. Step 3: Determine the other multiple to get your radicand with the perfect square. Step 4: Simplify your perfect square. Leave the other number out. Step 5: Place the two together for your final answer.

Unsimplifying Square Roots:

Step 1: Raise your perfect square to an exponent of 2 and simplify. Step 2: Multiply the raised number and radicand together. Step 3: You have your unsimplified root.

Note:

These processes also work for fifths, sixths, sevenths, etc.

First thing is first:

You might want to know some perfect squares, cubes, and fourths.

Perfect Cubes - Through 13:

³√1 = 1 ³√8 = 2 ³√27 = 3 ³√64 = 4 ³√125 = 5 ³√216 = 6 ³√343 = 7 ³√512 = 8 ³√729 = 9 ³√1000 = 10 ³√1331 = 11 ³√1728 = 12 ³√2197 = 13

Example: Cube Root.

³√375 Perfect Cube that goes into 375: 125. ³√125 × 3 ³√125 = 5 5³√3 (note the ³ sticks with the radicand of 3.)

Perfect Fourths - Through 13

⁴√1 = 1 ⁴√16 = 2 ⁴√81 = 3 ⁴√256 = 4 ⁴√625 = 5 ⁴√1296 = 6 ⁴√2401 = 7 ⁴√4096 = 8 ⁴√6561 = 9 ⁴√10000 = 10 ⁴√14641 = 11 ⁴√20736 = 12 ⁴√28561 = 13

Example: Fourth Root.

⁴√162 Perfect Fourth: 81 ⁴√81 × 2 ⁴√81 = 3 3⁴√2 (Like the last problem, the ⁴ is with the radicand, 2).

Perfect Squares - Through 13:

√1 = 1 √4 = 2 √9 = 3 √16 = 4 √25 = 5 √36 = 6 √49 = 7 √64 = 8 √81 = 9 √100 = 10 √121 = 11 √144 = 12 √169 = 13

Example 2

√32 Factors (other than 1 and 32): 2, 4, 8, 16 Any perfect squares? Yes, there are TWO, 4 and 16. What to do? Let's try using 4 first. 4x = 32 x = 8 √4 × 8 √4 = 2 2√8 Would this be an acceptable answer? Answer: No. Why? 8 can still be simplified because it contains the perfect square of 4 as one of its factors. Then we would go 2√ 4 × 2 √4 = 2 2 + 2√2 4√2 (by the way, this is what you get when you use 16. I will avoid that work.)

Simplifying a Square Root - Example:

√75 Factors (other than 1 and 75): 3, 5, 15, 25 Any perfect squares? Yes, 25. 25x = 75 x = 3 √25 × 3 √25 = 5 Answer: 5√3

Now that we have listed the basic perfect squares, cubes and fourths, we can get to simplifying.

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Example - Fourth Root:

2⁴√4 Raise 2 to an exponent of 4. 2⁴ = 16 Multiply. 16 × 4 = 64 Answer: ⁴√64

Example

3√4 Raise 3 to an exponent of 2. 3² = 9 Multiply. 9 × 4 = 36 Answer: √36

Example- Cube Root:

4³√5 Raise four to an exponent of 3. 4³ = 64 Multiply. 64 × 5 = 320 Answer: ³√320

Thanks!

Created on 3/16/2017 by CA7the2nd, hope this was helpful.

Unsimplifying Cubes and Fourths:

Like square roots except to a power of 3 or 4.

Simplifying Cube Roots and Fourth Roots:

Process is the same, results may look slightly different.

Hi everyone, just want to explain how to simplify and unsimplify different roots, give some examples, and hopefully any doubt or question you had in it before will be cleansed and gone, forevermore.

Ready?