

PDF Worksheet and Answer Key
Read through the lesson and then give it a shot! 
The "radical" ( \( \sqrt{ \ } \) ) is likely the first daunting squiggle one encounters in basic math. Although it may seem strange, the radical just poses the following question:
\( \sqrt[n]{m} \) asks, what number (or numbers) can I raise to the \( n^{th} \) power to get \( m \)? Let's try a few out. If you're not feeling good about exponential "powers", try the lesson first. 
TERMS: In the example to the left, "n" is called the 'index of the radical', and "m" is called the 'radicand' NOTE: When there is no index (n) written, it is an assumed "2". 
\( \sqrt[3]{8} \) or "What numbers can I raise to the 3rd power to get 8?"

We can check that \( 2^3 = 8 \), so \( \sqrt[3]{8} \) = 2

\( \sqrt{16} \) or "What numbers can I raise to the 2nd power to get 16?"

We can check that \( 4^2 = 16 \), so \( \sqrt{16} \) = 4
But hold up, because \( (4)^2 = 16 \) as well! 
That's right, there were two answers to that last one.
In fact, there are two answers to any root of an even index, (unless the radicand is zero, that is)
while there is only one answer to any root of an odd index.
Luckily for us, in the roots of even index, the answers will always be the positive and negative of the same number!
The positive result is called the principle root.
In fact, there are two answers to any root of an even index, (unless the radicand is zero, that is)
while there is only one answer to any root of an odd index.
Luckily for us, in the roots of even index, the answers will always be the positive and negative of the same number!
The positive result is called the principle root.
OK, that's all well and good, but can I take a root of ANY number?
Well, most of the time roots require approximation by your handy dandy calculator. (Example: \( \sqrt{45.3} \) is around 6.7305)
However, think about the question posed above. "What number(s) can I raise to the index power to get the radicand"
If the index is even, then I have to raise a number to an even power, and that result will always be positive. So, with an even index, I can't have a negative radicand! Let's look at one to get a good feel:
However, think about the question posed above. "What number(s) can I raise to the index power to get the radicand"
If the index is even, then I have to raise a number to an even power, and that result will always be positive. So, with an even index, I can't have a negative radicand! Let's look at one to get a good feel:
\( \sqrt{16} \) or "What numbers can I raise to the 2nd power to get 16?"

This is impossible with real numbers and requires complex numbers (often poorly labeled as imaginary numbers). These numbers will be tackled in a later section!
Since we are only working with real numbers in this section, our 'answer' in this case would be UNDEFINED. 
\( \sqrt[3]{8} \) or "What numbers can I raise to the 3rd power to get 8?"

This is possible, since we are raising to an odd power.
We can check that \( (2)^3 \) is indeed 8! 