How to simplify square root fractions?

To simplify fractions with square roots you rationalize the denominator.

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"Simplifying" radicals is actually just writing the radical in an equivalent standard form. Often the simplified version seems more complicated. Take, for instance, `sqrt(18)=3sqrt(2)` . The expression on the right, while simplified, requires two operations (multiplication and root extraction) while the expression on the left just requires finding the root.

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"Simplifying" radicals is actually just writing the radical in an equivalent standard form. Often the simplified version seems more complicated. Take, for instance, `sqrt(18)=3sqrt(2)` . The expression on the right, while simplified, requires two operations (multiplication and root extraction) while the expression on the left just requires finding the root.

There are a couple of reasons to write in simplified form. One is that, done correctly, everyone has the same expression. This makes it easier to discuss (or grade.) Another reason is that the simplified form can sometimes highlight a pattern. The diagonals of squares with integral lengths are `sqrt(2), sqrt(8), sqrt(18), sqrt(32),...` There is a pattern in the radicands but the pattern is quite obvious when written in standard form: `1sqrt(2),2sqrt(2),3sqrt(2),4sqrt(2),...`

A square root is simplified if there are no perfect square factors in the radicand, there are no fractions in the radicand, and there are no fractions in the denominator.

We can always fix problems with the second rule by using the property `sqrt(a/b)=sqrt(a)/sqrt(b);b ne 0` . Then `sqrt(3/4)=sqrt(3)/sqrt(4)=sqrt(3)/2`.

The real difficulty is with the third rule. `sqrt(3/7)=sqrt(3)/sqrt(7)` . Now, there are no perfect square factors in either radicand and there are no square roots in the radicands. However, we have introduced a square root in the denominator. To eliminate this we rationalize the denominator.

This involves multiplying the expression by a fancy form of one such that when we multiply the denominators we get a rational number.

In the case `sqrt(3)/(sqrt(7)` we can multiply by `sqrt(7)/sqrt(7)` to get `sqrt(21)/7` which is the simplified form of `sqrt(3/7)` . (Note that a calculator returns the same value for each approximately 0.6546536707.)

We can run afoul of the other rules so you must be careful. For example `sqrt(5/8)=sqrt(5)/sqrt(8)=sqrt(5)/sqrt(8) * sqrt(8)/sqrt(8)=sqrt(40)/8` But 40 has a perfect square factor of 4 so `sqrt(40)/8=(2sqrt(10))/8=sqrt(10)/4` . One way to avoid this is to simplify both numerator and denominator before you rationalize.

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