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

Evaluate

\[ ( \sqrt{ 12 } - 1 ) ( \sqrt{ 12 } + 1 ). \]

True or False:

For all real numbers \( a \) and \( b, \)

\[ \left( \sqrt{a} + \sqrt{b} \right) ^2 = a + b .\]

How many real solutions are there to

\[ x - 4 = 2 \sqrt{ x - 1 } ?\]

The velocity (V) of an object can be obtained from the kinetic energy (KE) of the object and its mass (M) using the following formula:

\[ V = \sqrt{ \frac{ 2 KE } { M } }. \]

If the new kinetic energy of the object (with the same mass) is 4 times the old kinetic energy, what can we say about the new velocity of the object?

What is the value of

\[ \sqrt{ 1 + \frac{ 16}{ 9 } } ? \]

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