Multiplying radicals

The radical sign was said to be developed in 1500s in Germany. The radical ( sign was invented to find the root of a number, it can be a square root or the cube root, 4th root, 5th root etc., depending on the index of the radical sign.

Radicals can be evaluated and simplified like in the case of and they can also be converted from an entire radical, to a mixed radical or vice versa like in the case of . In converting radicals we need to apply our skills in factoring the perfect cubes or the perfect squares inside the radical sign.

We will also be learning how to evaluate radicals without the need to use calculator. There are a lot of times when there are fractions inside the radical, and there would also be times when there would be decimals. We will learn how to manually calculate them and also use a free radical calculator to check our answers.

There are also times when we will encounter the two kinds of radicals, which are the entire radicals like and the mixed radicals like. We would also be learning how to convert them into either entire radical to mixed radical or vice versa, by applying our knowledge of factoring out the perfect squares or the perfect cubes in the equation.

Apart from converting the radicals from one form to another we are also going to learn more about how to combine different radicals and simplify them depending on the operation. For this chapter we would be focusing a lot in adding, subtracting and multiplying radicals. For addition and subtraction, the rule is simply to combine like radicals. Like radicals are the radicals that have the same radicand. In multiplying radicals, if we’re given entire radicals, we just need to multiply the radicands. If we have mixed radicals then we proceed with multiplying the coefficients first, and then multiplying the radicand. If there is a need to simplify the radicand, then we must simplify.

Each section of this chapter would discuss the topics mentioned above. It would be very useful to review the past concepts that we discussed about factoring, and radicals.

Multiplying radicals

It requires 2 steps to multiply radicals. First is to multiply the numbers inside the radical sign, the radicands, together. Second is to multiply the numbers outside the radical sign together. Don’t forget that only radicals with the same index can be combined through multiplication!


  • 2.
    Multiplying Radicals
    Multiply and simplify.
    • b)
      (28)(518)\left( {2\sqrt 8 } \right)\left( {5\sqrt {18} } \right)
    • c)
      (5x)(6y)\left( {5\sqrt x } \right)\left( {6\sqrt y } \right)
    • d)
      33\sqrt {3} \cdot \sqrt {3}
  • 3.
    Expand and simplify.
    • a)
      3(563)\sqrt 3 \;\left( {5\sqrt 6 - \sqrt 3 } \right)
    • c)
      3(27)3(1221)3\;\left( {\sqrt 2 - \sqrt 7 } \right) - \sqrt 3 \left( {\sqrt {12} -\sqrt {21} } \right)
    • d)
      3x(x5y)-3\sqrt x\;\left( {\sqrt x -5 \sqrt y } \right)
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Multiplying radicals

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