The first law of Radicals

First law of radicals

Consider the expression \sqrt{4.9}. We know that \sqrt{4.9}=\sqrt{36}=6. Also, \sqrt{4}.\sqrt{9}=2.3=6. Therefore we can write ;

\sqrt{4.9}=\sqrt{4}.\sqrt{9}.

Also Read:

This example illustrates an important property of radicals. The property states that a radical sign may be distributed over a product provided that each factor is positive, and it is true for every index n.

First law of radicals

\sqrt[n]{ab}=\sqrt[n]{a}.\sqrt[n]{b}

For all positive number a and b.

The first law of radicals can be proved as follows:

Let x=\sqrt[n]{a}\text{ and }y=\sqrt[n]{b}. Then x is an nth root of a and y is an nth root of b. By the definition of nth root we can write that x^n=a\text{ and }y^n=b. Therefore we have :

\sqrt[n]{a}.\sqrt[n]{b}=x.y=\sqrt[n]{(xy)^n}=\sqrt[n]{x^ny^n}=\sqrt[n]{ab}

Which is the desired result.

We may use the first law of radicals to simplify certain types of radical expressions. For example, to simplify \sqrt{75} we find the largest factor of 75 which is perfect square and then apply the first law of radicals. Our work looks like this :

\sqrt{75}=\sqrt{25.3}=\sqrt{25}.\sqrt{3}=5\sqrt{3}

Notice that 5.\sqrt{3}=5\sqrt{3}

Example 1 # :

Simplify \sqrt{700}

Solution :

\sqrt{700}=\sqrt{100.7}=\sqrt{100}.\sqrt{7}=10\sqrt{7}

Using this last form it is an easy matter to approximate \sqrt{700}. The number 700 is too large to be found in our table of square roots, so we find the square root of 7 and multiply by 10.

\sqrt{700}=10\sqrt{7}=10.(2.646)=26.46

Also Read : roots and radicals

Example 2 # :

Simplify 5\sqrt{18}

Solution :

5\sqrt{18}=5\sqrt{9.2}

5\sqrt{18}=5\sqrt{9}.\sqrt{2}

5\sqrt{18}=5.3\sqrt{2}

5\sqrt{18}=15\sqrt{2}

Example 3 # :

Simplify \sqrt[3]{250}

Solution :

In this case we look for the largest perfect cube that is a factor of 250. The procedure that was used to factor large integers. This procedure is illustrated below :

250 = 2. 5. 5. 5

Therefore,

\sqrt[3]{250}=\sqrt[3]{2.5^3}

\sqrt[3]{250}=\sqrt[3]{2}.\sqrt[3]{5^3}

\sqrt[3]{250}=5\sqrt[3]{2}

Example 4# :

Simplify \sqrt{288x^5}

Solution :

288 = 2. 2. 2. 2. 2. 3. 3

x^5=x^4.x

\sqrt{288x^5}=\sqrt{2^4.2.3^2.x^4.x}

=\sqrt{2^4.3^2.x^4}.\sqrt{2.x}

=2^2.3.x^2.\sqrt{2.x}

=12x^2\sqrt{2x}

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