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How Do You Multiply Two Radicals?

Multiply: 2 √3xz • √6xy3

Summary

  1. The Product Property of Square Roots helps you multiply radicals together or break them apart
  2. The symbol above '3xz' and '6xy3' is called a square root
  3. Perfect squares are values that can be written as something squared
  4. The Product Property also lets us split our big radical into several smaller radicals multiplied together
  5. The vertical lines around 'x' mean absolute value

Notes

    1. A radical is another name for a square root
    1. A radical is another name for a square root
    2. So that means we can multiply 3xz and 6xy3 together and put it under one big square root
    1. Multiply constants together and common variables together
    2. So we multiply 3•6 to get 18
    3. Then we multiply x•x to get x2
    4. Don't forget the y3 and z at the end!
    1. We still have some perfect square factors under our radical, so it's not in simplest radical form
    2. We need to factor out those perfect squares in order to simplify
    1. Look for factors that are perfect squares
    2. Perfect squares are values that can be written as something squared
    3. We can take the square root of a perfect square and get a whole number as the answer
    4. Variables with even exponents are also perfect squares
    1. We can factor 18 into 9•2
    2. 9 is a perfect square: it can be written as 32
    3. x2 is also a perfect square: it's already written as something squared
    4. y3 can be rewritten as y2•y
    5. y2 is a perfect square
    1. Before we used the Product Property to combine multiple radicals into one
    2. But we can also use it the other way to break up one big radical into several radicals
    1. The Product Property of Square Roots lets you break up a radical into multiple radicals being multiplied together
    1. Now we just need to simplify some square roots!
    1. 9 = 32, so the square root of 9 is 3
    2. 'x' started with an even exponent and ended up with an odd exponent
    3. So we need to put absolute value bars around 'x'
    4. Since 'y' started with an odd exponent, we don't need to worry about absolute value symbols
    1. Now we don't have any more perfect square factors under the square root
    2. So our expression is in simplest form!