Topic: Higher Roots (corresponds to Session 4, roughly, in the textbook)
I elaborated on the Math is Fun page on n-th roots. My comments and summary are below, but please read that page also and follow the link to “fractional exponents” to get more depth.
Definitions: I will use exponential notation rather than the multiplicative notation that is used in the Math is Fun page
In what follows, $n$ represents a natural number greater than 1, and $a$ represents a real number.
An n-th root of $a$ is a number, $b$, such that $b^{n} = a$
I’m not using the notation $\sqrt[n]{a}$ in that definition, because of this
Note: (not mentioned in Math is Fun, but I mentioned) There may be more than one n-th root of $a$ in the real numbers, in which case the notation $\sqrt[n]{a}$ refers to the positive n-th root.
However, it is a general fact that
$\left(\sqrt[n]{a}\right)^{n} = a$
this is another way to say the definition of square root.
Properties of n-th roots:
Roots and multiplication:
(If n is even, a and b must both be ≥ 0)
Roots and division:
(If n is even, then a≥0)(and b>0 no matter what n is)
(b cannot be zero, as we can’t divide by zero)
Roots and addition/subtraction: they don’t work well together!
Watch out for the following:
$\sqrt[n]{a + b} \neq \sqrt[n]{a} + \sqrt[n]{b}$
$\sqrt[n]{a – b} \neq \sqrt[n]{a} – \sqrt[n]{b}$
$\sqrt[n]{a^{n}+ b^{n}} \neq {a} + {b}$
$\sqrt[n]{a^{n} – b^{n}} \neq {a} – {b}$
Roots of powers:
Here’s the handy table from the Math is Fun page:
| n is odd | n is even | |
|---|---|---|
| a ≥ 0 |  |
|
| a < 0 | |
 |
I discussed at some length why there needs to be an absolute value in that bottom right cell of the table.
Quiz next time: Explain why we need the absolute value in the rule $\sqrt{x} = |x|$. Why do we not need an absolute value in the rule $\sqrt[3]{x} = x$?
More Roots of Powers:
A useful property: $\sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m}$ for a ≥ 0
Sometimes this is helpful in simplifying n-th roots. But even more powerful is to use fractional exponents to represent roots:
Definitions:
$\sqrt[n]{a} := a^{\frac{1}{n}}$ for a ≥ 0
$\sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m} := a^{\frac{m}{n}}$ for a ≥ 0
Homework homework homework yes there is homework!
Before Thursday’s class, please do the following:
• Read and try to understand the notes above, notes you took in class, the Math is Fun page on n-th roots, and also the page on fractional exponents.
• Do the “Your Turn” problems at the bottom of the Math is Fun page on n-th roots
• Do the WeBWorK “HigherRoots”
• Start working on the WeBWorK “HigherRootsAlgebraic” – we will probably pick up on this tomorrow.
