document.write( "Question 622589: Solve $\"sqrt%28x-5%29\"$ =cube root(x-3) \n" ); document.write( "

Algebra.Com's Answer #391561 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
If there is no error in what you posted, then the best I can do for you is to show you how to eliminate the radicals. The problem you posted is not easy to solve. If there is an error in what you posted then maybe what you see below will help you figure out how to do the problem on your own.

\n" ); document.write( " $\"sqrt%28x-5%29=root%283%2C+x-3%29\"$
\n" ); document.write( "This equation has two kinds of roots, square and cube. TO eliminate the radicals we can either do them one at a time or both together. To eliminate them both with one operation it will help to rewrite them with fractional exponents instead of radicals:
\n" ); document.write( " $\"%28x-5%29%5E%281%2F2%29=%28x-3%29%5E%281%2F3%29\"$
\n" ); document.write( "Now we raise both sides to the lowest common denominator (LCM) power. (You'll see why shortly.) The LCM of 2 and 3 is 6:
\n" ); document.write( " $\"%28%28x-5%29%5E%281%2F2%29%29%5E6=%28%28x-3%29%5E%281%2F3%29%29%5E6\"$
\n" ); document.write( "The rule for exponents when raising a power to a power is to multiply the exponents. So this simplifies to:
\n" ); document.write( " $\"%28x-5%29%5E3=%28x-3%29%5E2\"$
\n" ); document.write( "Now that the radicals are gone, the equation becomes solvable (in theory). As usual we start the solution by simplifying. Cubing the left side, (x-5)(x-5)(x-5), and squaring the right side, (x-3)(x-3), we get:
\n" ); document.write( " $\"x%5E3-15x%5E2%2B75x-125=x%5E2-6x%2B9\"$
\n" ); document.write( "With these exponents this is not a linear equation. So we want one side to be zero. Subtracting the entire right side from both sides of the equation we get:
\n" ); document.write( " $\"x%5E3-16x%5E2%2B81x-134=0\"$
\n" ); document.write( "Next we would factor. Then we would use the Zero Product Property to solve the equation. But this does not factor. At least one solution must be irrational. \n" ); document.write( "

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