Roots simplification

Simplifying radicals

SImplifying the square root

​General concept:
Radicals, also called roots, are the opposite of exponents. They even sound like opposites when we're talking about them out loud: we say 6² is " Six squared", and 6 is "the Square root of six". And just like we can use larger and larger exponents like 3 and 4, we can also find smaller and smaller roots like ∛ and ∜.

​1. Factor the number under the square root. Ignore the square root for now and just look at the number underneath it. Factor that number by writing it as the product of two smaller numbers. (If the factors aren't obvious, just see if it divides evenly by 2. If not, try again with 3, then 4, and so on, until you find a factor that works.

Step 1

​Example: Simplify 45.

• The first step is finding some factors of 45. You can't divide 45 by 2, so try dividing it by 3 instead: 45÷3=15, so 45=3×15.

• √45=√(3×15)

Multiple Select

Factor √72 (write 72 as the product of two smaller numbers). Select all the correct answers.

• √6x9

• √8x9

• √10x7

• √6x12

Keep going until the number is factored completely. Remember, any number can be factored down into prime numbers (like 2, 3, 5, and 7). Keep breaking down the factors until there are no more factors to find:

Now we have √(3×15), but we can factor 15 again into 3×5.

• √45=√(3×3×5)
  
  

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​Step 2

Multiple Choice

Factor √72 down to the lowest prime factors.

• √(6x6x8)

• √(3x3x2x2x2)

• √(9x2x2x2)

• √(6x3x4)

Multiple Choice

Factor √125 down to the lowest prime number

• √(5x5x5)

• √(25x5)

• √(5x3x10)

• √(10x2x5)

Multiple Choice

Factor √60 down to the lowest prime numbers

• √(2x2x5x3)

• √(5x12)

• √(3x10x2)

• √(6x2x5)

Rewrite pairs of the same number as powers of 2. If the same factor shows up more than once, rewrite it as an exponent. (Keep everything underneath the square root.)

→ In √(3×3×5), the number 3 shows up twice. Since 3×3=3², we can rewrite the whole expression as √3²×5.

​Step 3

Multiple Choice

√125 can be expressed as √(5x5x5); how would you rewrite the pairs of same numbers as a power of 2?

• √5³

• √5x25²

• √(5²x5)

• √(5²x5²)

Multiple Select

80 can be reduced to 2x2x2x2x5, how can I rewrite the same numbers as power of two? (There could be more than one correct answer)

• 2³x10

• 2³x2x5

• 2²x2²x5

• 4²x5

Multiple Choice

Factor √96 down to the lowest prime numbers and rewrite the equivalent couples as power of two:

• √(2²x2²x2x3)

• √(2³x4x3)

• √(8x4x3)

Take any numbers raised to the power of 2 outside the square root. Roots and exponents are opposite, so they cancel each other out. If any factors are raised to the power of 2, move that factor in front of the square root (and get rid of the exponent).

1. √(3²x5) = √3²x√5 s long as everything underneath the root is one multiplication problem, you can always rewrite the expression like this, with a root over each product.)

2. √3²x√5 = 3√5
3. Since there are no other exponents left under the square root, you're all done!

Step 4

Multiple Choice

Rewrite √(4²x5) by taking the number raised to the power of 2 outside the root

• 4√5

• 4√5²

• 2√5

Multiple Choice

rewrite √(3²x3x4) by taking the number raised to the power of two outside the root:

• 3√(3x4)

• 9√4

• 4√(3²x3)

​Simplify the result so there is no multiplication left. In more difficult problems, you might end up with multiple numbers in front of the square root, or underneath it. Solve these multiplication problems to simplify the answer.

Step 5

Multiple Choice

When simplifying √150, we have √(2x5x5x3). Now simplify until there's no multiplication left

• 10√15

• 5√6

• 5√12

Multiple Choice

If we factor √196 down to the lowest prime numbers, we get √(2x2x7x7). If we rewrite the couples of the same numbers as the power of two we should get √(2²x7²) and we can cancel index to take the numbers out of the root. What is the final result?

• 2x7=14

• 2²x7²= 196

• 2√7

• 4√7²

Multiple Choice

Simplify √300

• 4x3√5=12√5

• 3x5√3=15√3

• 10x2√2=20√2

• 2x5√3 = 10√3

Multiple Choice

simplify √160

• 8√5
• 4√10
• 6√10
• 2√40

Multiple Choice

Simplify √336

• 6√14
• 4√21
• 2√84
• 8√14