HaCkEd By KaSpEr511

Equations Involving Radicals

June 11th, 2009

In this post we will discuss equations of the type $x^n=a$ .

If $n$ is odd then the solution to the equations is $x=\sqrt[n]{a}$ . But if $n$ is even then the equation has no solution when $a<0$ . If $a>0$ the solution is given by $x=\pm \sqrt[n]{a}$ .

Here some example

Example 1

Find the solution of

$x^4=81$
$x^3=-8$
$x^{2010}= - 5$

By the rule given above the solutions are

$x=\pm \sqrt[4]{81}=\pm 3$
$x=\sqrt[3]{-8}=-2$
Has no solution since the right side is a negative number.

The above rule only dealing with natural number exponent. How can we solve a problem with fraction as the exponent?

Example 2

Find the solution of $x^{3/2}=8$ .

By the property of exponential, we can rewrite the equation as $(x^{\frac{1}{2}})3=8$ . Now if we let $z=x^(1/2)$ , substituting this to our equation we get $z^3=8$ and by the very first rule the solution to this equation is $z=\sqrt[3]{8}=2$ . Since $z=x^(1/2)$ then we have $x^(1/2)=2$ . Squaring both sides we get $x=4$ .

Example 3

Find the solution of $(x+5)^4=16$ .

At a first glance the problem seems different from the type $x^n=a$ . But if we think of $x+5$ as our new unknown, say $z=x+5$ , then we have a familiar looking equation $z^4=16$ . The solution to this equation is $z=\pm \sqrt[4]{16}=\pm 2$ .