01Dsimplifying

toc =** Simplifying Surds **=

Recall that we can simplify some fractions to make them easier to work with.

math \dfrac{8}{16} = \dfrac{1}{2} math

In the same way, we can simplify some surds.

Recall that, if the surd contains a perfect square (like 16), the surd simplifies to an integer : math \sqrt{16} = 4 math

Multiplying surds
We will use the idea that when you multiply two surds, just multiply the insides together.

math \text{Eg : } \sqrt{5} \times \sqrt{2} = \sqrt{5 \times 2} = \sqrt{10} math

The reverse of this is used to break a surd into two factors

math \text{Eg : } \sqrt{15} = \sqrt{5} \times \sqrt{3} math

If we make sure one of the factors is a perfect square, the perfect square factor will simplify to an integer.

math \\ \text{Eg : } \sqrt{20} = \sqrt{4} \times \sqrt{5} \\ \\ . \qquad \qquad = 2 \times \sqrt{5} \\ \\ . \qquad \qquad = 2\,\sqrt{5} math

{as with all algebra, we don't write the "×" sign, just push the two things together, like //**a × b = ab**//}

Perfect Squares
A ** perfect square ** is the result of squaring an integer.

I find it useful to have a list of the perfect squares available to refer to :

{start with the first 12 or so integers and square each one -- 1 is also a perfect square but ignore it} {the perfect squares are the numbers in the second row}

Simplifying Surds
To simplify any surd:
 * 1) Find a factor which is a perfect square (from the __second__ row of the above table)
 * 2) break the surds into a product of two factors
 * 3) simplify the surd which is a perfect square to an integer

** Example 1 **
math \textbf{(a)} \quad \text{Simplify } \sqrt{75} math

{From our list, 25 is a perfect square and a factor of 75, so break 75 into 25 × 3} math \\ . \qquad \sqrt{75} \; = \sqrt{25} \times \sqrt{3} \\ \\ . \qquad \qquad \; = 5 \times \sqrt{3} \\ \\ . \qquad \qquad \; = 5 \, \sqrt{3} math

math \textbf{(b)} \quad \text{Simplify } \sqrt{200} math

{from our list, 100 is a perfect square and a factor of 200, so:} math \\ . \qquad \sqrt{200} \; = \sqrt{100} \times \sqrt{2} \\ \\ . \qquad \qquad \; \; = 10 \, \sqrt{2} math

math \textbf{(c)} \quad \text{Simplify } 5 \, \sqrt{72} math

math \\ . \qquad 5 \, \sqrt{72} \; = 5 \times \sqrt{36} \times \sqrt{2} \\ \\ . \qquad \qquad \quad = 5 \times 6 \, \sqrt{2} \\ \\ . \qquad \qquad \quad = 30 \, \sqrt{2} math

Entire Surds
An ** entire surd ** is a surd with no extra value out the front -- so the entire value is __inside__ the square root sign.

math \text{An } \textbf{entire surd} \text{ is in the form } \sqrt{a} math

To change a mixed surd back into an entire surd, square the outside number and multiply it into the surd.

** Example 2 **
math \textbf{(a)} \quad \text{Write } 5 \, \sqrt{3} \text{ as an entire surd } math

math \\ . \qquad 5 \, \sqrt{3}\; = \sqrt{25} \times \sqrt{3} \\ \\ . \qquad \qquad \; \; = \sqrt{75} math

math \textbf{(b)} \quad \text{Write } 4 \, \sqrt{7} \text{ in the form } \sqrt{a} math

math \\ . \qquad 4 \, \sqrt{7} \; = \sqrt{16} \times \sqrt{7} \\ \\ . \qquad \qquad \;\; = \sqrt{112} math

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