Multiply and Divide Square Roots

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SWBAT multiply, divide, and simplify expressions involving square roots.

Big Idea

Similar to powers, square roots distribute over products and quotients. This is a key lesson for fluency in operating with square roots.


12 minutes

Hand each student a copy of Launch Mult and Div SqRts.docx to complete. Pairs work well here because one of the students can use the calculator while the other records. The roles should be switched for the next section when division of square roots is addressed. This task helps students think of square roots of numbers by exposing them to their decimal approximations. Many students view the radical sign as something separate from the radicand. Walk around to make sure the students are actually writing the decimals. 


20 minutes

In the launch task, students should have found that when the product of two numbers a and b is a third number (ab), then the product of the square root of a and the square root of b is the square root of ab.

Write on the board: a · √b = √ab , for all positive real numbers.

Tell students that this is The Product of Square Roots Property.

At this point show students that this is a special case of the Power of a Product Property, with the exponent = ½

Address the class and ask students to:

  • Find ?: √x = x?
  • Simplify (ab)1/2 using power of a product.
  • Ask the students if they can demonstrate the connection between the Square Root property and Power of a Product.

Students should state that the Square Roots Property is really the Power of a Product when the exponent is ½

Finalize the discussion by writing on the board:

 √a · √b = √ab  is the same as  a1/2 ∙ a1/2 = (ab)1/2


After the discussion, project the New Info task.docx on the board and ask students to follow the instructions. 

Students should conclude Quotient of Square Roots Property in question 2 is really the Power of Quotient property when the exponent is 1/2.  






15 minutes

Hand each student a copy of Application Worksheet.docx and ask them to work independently, at first. Then, ask students to work in pairs to check each other's results. 

In question #2, students may multiply the coefficients by the radicand. (3√3 as √9, for example) If this occurs, ask the students to verify if the statement is true with their calculators. Indicate that the coefficients and square root expressions must be multiplied separately. 

Call on student volunteers to go up to the board and write their answers so that the class can review the correct answers.


5 minutes

To close the lesson, ask students use the back of their Application Task Sheet and answer the following question. 

Given the expression √75

Write this expression as the product of two powers and also as the quotient of two powers. 

If time permits, call on a few students to share their answers out loud....product of two powers first, then quotient of powers. 


The homework assignment is short yet a proper continuation of the lesson