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Lesson 6 of 12

    Applying the underlying structure of multiplication and division of surds

    I can multiply and divide with surds.

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    Lesson 6 of 12
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      Applying the underlying structure of multiplication and division of surds

      I can multiply and divide with surds.

      Download all

      Lesson details

      Key learning points

      1. The multiplication of surds can be generalised.
      2. √a × √b=√ab and √a × √(1/b) = √(a/b) = √a ÷ √b
      3. You may be able to simplify this product.

      Keywords

      • Surd - A surd is an irrational number expressed as the root of a rational number.

      • Radical - The root sign is the radical symbol.

      • Radicand - The radicand is the value inside the radical symbol.

      Common misconception

      Pupils may insist that a square root can be both positive and negative.

      By convention, square root refers to the principal (positive) square root.

      For √x to be a function, it can only have one output for each input. Graphing this using Desmos is a great way to demonstrate this and more on this will be covered in the functions and proof unit.
      Teacher tip

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      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

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      6 Questions

      Q1.
      Calculate $$\sqrt {0.764} \times \sqrt {0.764}$$

      Correct Answer: 0.764

      Q2.
      Calculate $${9 \over \sqrt {3}} \times {9 \over \sqrt {3}}$$

      Correct Answer: 27

      Q3.
      $$2\sqrt {5} \times \sqrt {2} = 2\sqrt {b}$$. What is the value of $$b$$?

      Correct Answer: 10, ten

      Q4.
      $$\sqrt {6} \times \sqrt {4} \times 3\sqrt {10}\times \sqrt {10} = a\sqrt {6}$$. What is the value of $$a$$?

      Correct Answer: 60, sixty

      Q5.
      Simplify the following expression: $$\sqrt {24} \div \sqrt {6}$$

      Correct Answer: 2, two

      Q6.
      Evaluate the following expression: $$3\sqrt {20} \div \sqrt {5}$$

      Correct Answer: 6, six

      Assessment exit quiz

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      6 Questions

      Q1.
      Simplify the expression $$\sqrt {4ab} \times \sqrt {7a^{2}b}$$

      $$2\sqrt {ab} \times \sqrt {7a^{2}b}$$
      $$2\sqrt {ab} \times a\sqrt {7ab}$$
      $$\sqrt {28a^{3}b^{2}}$$
      Correct answer: $$2ab\sqrt {7a}$$

      Q2.
      Simplify the expression $$\sqrt {3 \over 4} \times \sqrt {1 \over 3}$$

      Correct answer: $$1 \over 2$$
      $$\sqrt {3 \over 12}$$
      $$1 \over 4$$
      $$\sqrt {1 \over 4}$$

      Q3.
      Simplify the expression: $$\sqrt {12} \times \sqrt {27}$$

      $$\sqrt {324}$$
      Correct answer: 18
      $$2\sqrt {3} \times 3\sqrt {3}$$
      54

      Q4.
      Simplify $$\sqrt {48} \times \sqrt {12}$$

      Correct Answer: 24, twenty-four, twenty four

      Q5.
      Calculate $$(2\sqrt {48} - 5\sqrt {48}) \times \sqrt {12}$$

      Correct Answer: -72

      Q6.
      Complete the rule for dividing surds: $$\sqrt {x \over y} = $$

      $$\sqrt {x} \over y$$
      $$x \over \sqrt {y}$$
      Correct answer: $$\sqrt {x} \over \sqrt {y}$$

      Lesson appears in

      UnitMaths / Surds

      Maths

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