chapter9.htm

The following symbol, $\sqrt{}$ , is called a "radical" symbol or a "root" symbol. In general, you'll see a number inside this little "v" portion of the radical (root) symbol. That number inside the v portion of the radical (root) symbol is called an "index." The math expression "inside" the radical symbol is called the "radicand."

A radical with an index of 2, $\root{2} \of{}$ is called a "square root." In general, math people do not place a 2 in the index position: if you see $\sqrt{}$ it is understood to be a "square root." A radical with an index of 3, $\root{3} \of{}$ is called a "cube root." A radical with an index of 4, $\root{4} \of{}$ is called a "fourth root." A radical with an index of n, $\root{n} \of{}$ is called a "nth root."

ROOTS WITH INDEXES THAT ARE POSITIVE EVEN INTEGERS

If "a" is a positive even integer and "b" is a positive real number

then $\root{a} \of{b}$ means

"What positive real number raised to the power of "a" will result in "b"?"

ex. 1) √ 4 is asking "What positive real number raised to the power of 2 will result in 4?"

The answer is 2.
Why? Because 2² = 4. Hence, we say √ 4 = 2.

Comment: Rememeber if there is no index in the root, the root is assumed to have an index of 2.

ex. 2) is asking "What positive real number raised to the power of 4 will result in 81?"

The answer is 3.
Why? Because 3⁴ = 81. Hence, we say = 3.

ex. 3) is asking "What positive real number raised to the power of 6 will result in 64?"

The answer is 2.
Why? Because 2⁶ = 64. Hence, we say = 2.

ROOTS WITH INDEXES THAT ARE POSITIVE ODD INTEGERS

If "a" is a positive odd integer and "b" is any real number

then $\root{a} \of{b}$ means

"What real number raised to the power of "a" will result in "b"?"

Comment: In this case your answer will be positive or
negative depending on the radicand.

ex. 1) is asking "What real number raised to the power of 3 will result in 8?"

The answer is 2.
Why? Because 2³ = 8. Hence, we say = 2.

ex. 2) is asking "What real number raised to the power of 5 will result in -32?"

The answer is -2.
Why? Because (-2)⁵ = -32. Hence, we say = -2.

Comment: Roots with odd indexes can have radicands that are negative and answers that are negative.

There is rule that allows you to transform a radical into an exponential. It is as follows:

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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 9: ROOTS