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What is the Product Property of Square Roots?

What is the Product Property of Square Roots?

Summary

  1. We can rewrite 36 as the product of its factors: 4•9
  2. We can also rewrite 6 as the product of its factors: 2•3
  3. When we square 2 we get 4, so 2 is equal to the square root of 4
  4. When we square 3 we get 9, so 3 is equal to the square root of 9
  5. Since both √(4•9) and √(4)•√(9) equal 2•3, they must be equal to each other
  6. Taking the square root of 'ab' is the same as taking the square root of 'a' times the square root of 'b'
  7. 'a' and 'b' need to be greater than or equal to 0 because we can't take the square root of a negative number

Notes

    1. We'll start by taking the square root of a number, 36
    1. 36 is a perfect square, which means that its square root is an integer
    1. 36 factors into 4 times 9
    2. 6 factors into 2 times 3
    1. We can rewrite the factors of 6 using square roots
    2. When we square 2 we get 4, so 2 is the square root of 4
    3. When we square 3 we get 9, so 3 is the square root of 9
    4. The symbol above the 4 and 9 is the square root symbol
    1. The Transitive Property of Equality says that if we have two quantities equal to the same thing, then those two quantities must be equal to each other
    2. So since both of these square root expressions equal 2•3, they must also be equal to each other!
    1. 'a' and 'b' need to be greater than or equal to 0 because we can't take the square root of a negative number
    2. This property lets us break up products under radicals into products of separate radicals
    3. This is really helpful when we're trying to simplify square roots!
    4. A 'radical' is just another name for the square root symbol
    1. Since 54 factors into 6•9, we can write √(6•9) instead of √(54)
    2. Then the Product Property allows us to break our single radical into the product of two radicals
    3. Remember, a 'radical' is just another name for the square root symbol
    4. We could still factor 6 into 2•3, but since neither 2 or 3 are perfect squares we wouldn't be able to simplify any further
    5. So now our radical is fully simplified!
    1. We can break up variable expressions by factoring them, just like with numbers
    1. If we factor an x2 out of x3, we'll be able to rewrite it under its own radical and simplify
    2. The Product Property lets us split this into two radicals multiplied together
    3. The square root cancels out the square on x2, so we're just left with x√(x)