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Radical Expressions and Equations

Combine your skills of working with exponents and manipulating expressions for radical glory.

Radical Equations

Solve the equation \[ \sqrt{ x+ 15 } = 9. \]

Solve the equation

\[ \sqrt{ 6 x + 27 } = x. \]

Solve for \(x\):

\[ \sqrt{x+141} - \sqrt{ 5x+85} = 2 \]

The solution to the equation \[ 1 + \sqrt{1 - \sqrt{x^4-x^2}} = x \] can be written in the form \( \frac{a}{b} \) where \(a\) and \(b\) are coprime positive integers. Find \(a+b\).

If \[\frac{\sqrt{180+x}}{\sqrt{29}}=3,\] what is the value of \(x\)?

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