Roots and Radicals

Review of radical expressions and extraneous solutions

The Essentials

Radicals are the inverse of exponents. The 2nd root (square root) undoes the power of 2. Likewise an nth root undoes a power of n and vice versa:

\[ \sqrt{y} = x \] \[ (\sqrt{y})^2 = x^2 \] \[ y = x^2 \]

However, when using a radical to undo an exponent, the positive and negative results must be used:

\[ y^2 = x \] \[ \sqrt{y^2} = \pm \sqrt{x} \] \[ y = \pm \sqrt{x} \]

This only applies when the power of the exponent and radical is even:

\[ y^3 = x \] \[ y = \sqrt[3]{x} \]

Some rad properties:

\[ \sqrt[n]{a} = a^{1/n} \] \[ \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \] \[ \sqrt[m]{ \sqrt[n]{a} } = \sqrt[nm]{a} \] \[ \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \]

When solving an equation such as this by squaring both sides:

\[ \sqrt{2x + 7} = x + 2 \]

We get the solutions: \( x = 1 \) and \( x = −3 \). However if we plug these into the equation \( x = 1 \) gives us \( 9 = 9 \), but \( x = −3 \) gives us \( \sqrt{1} = -1 \) which isn’t true. This incorrect solution is called an extraneous solution. It occurs because \( x = −3 \) is a solution for \( - \sqrt{2x + 7} = x + 2 \), not \( \sqrt{2x + 7} = x + 2 \). When we squared the equation, we got rid of this possibe answer

Sometimes, they say you can’t take the square root of negative numbers, in these cases ’they’ just don’t want to deal with imaginary numbers. The letter i is the square root of −1. When there is a square root of a negative, it is usually written in terms of i:

\[ \sqrt{-49} = \sqrt{49} \sqrt{-1} = 7i \]

Example

Simplify the expression:

\[ 3\sqrt{45} + \sqrt{20} \] \[ 3 \sqrt{9} \sqrt{5} + \sqrt{4} \sqrt{5} \] \[ 9 \sqrt{5} + 2 \sqrt{5} \] \[ 11\sqrt{5} \]

Practice

Simplify this expressions:

  1. \( \sqrt{75} + 7\sqrt{12} - 9\sqrt{3} \)

Find the solutions of these equations. Find which of the solutions (if any) are extraneous.

  1. \( \sqrt{7 - 3x} = x - 3 \)
  2. \( x + 4 = \sqrt{15x + 6} \)

Solution:

  1. \( 10 \sqrt{3} \)
  2. \( x = 2, 1 \). \( x = 1 \) is an extraneous solution
  3. \( x = 2, 5 \).

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