The nth Root
etc!| 2 | √a × √a = a | The square root provided two times in a multiplication offers the initial value. You are watching: Simplify square root of 3 multiplied by the fifth root of 3. | ||
| 3 | 3√a × 3√a × 3√a = a | The cube root used three times in a multiplication offers the original value. | ||
| n | n√a × n√a × ... × n√a = a(n of them) | The nth root supplied n time in a multiplication gives the initial value. |
The nth source Symbol
This is the unique symbol that way "nth root", that is the "radical" prize (used because that square roots) v a small n to average nth root.
Using it
We might use the nth root in a question favor this:
Question: What is "n" in this equation?
n√625 = 5
Answer: i just take place to know that 625 = 54 , so the 4th source of 625 must be 5:
4√625 = 5
Why "Root" ... ?
 | When you view "root" think "I know the tree, but what is the source that developed it? " Example: in √9 = 3 the "tree" is 9 , and the root is 3 . |
Properties
Now we know what an nth root is, let united state look at some properties:
Multiplication and Division
We deserve to "pull apart" multiplications under the root sign favor this:
n√ab = n√a × n√b (Note: if n is even then a and also b have to both it is in ≥ 0)
This can assist us leveling equations in algebra, and also make part calculations easier:
Example:
3√128 = 3√64×2 = 3√64 × 3√2 = 43√2
so the cube root of 128 simplifies come 4 time the cube root of 2.
It likewise works because that division:
n√a/b = n√a / n√b (a≥0 and b>0)Note the b cannot be zero, together we can"t divide by zero
Addition and Subtraction
But us cannot carry out that kind of thing for enhancements or subtractions!
n√a + b ≠ n√a + n√b
n√a − b ≠ n√a − n√b
n√an + bn ≠ a + b
Example: Pythagoras" theorem says
 | a2 + b2 = c2 |
So we calculate c favor this:
c = √a2 + b2
Which is not the exact same as c = a + b , right?
It is basic trap to loss into, for this reason beware.
It also method that, unfortunately, additions and subtractions deserve to be hard to resolve when under a source sign.
Exponents vs Roots
An exponent top top one next of "=" have the right to be turned into a source on the other side the "=":
If an = b climate a = n√b
Note: once n is even then b have to be ≥ 0
nth source of a-to-the-nth-Power
When a value has an exponent the n and also we take the nth root us get the value back again ...
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... As soon as a is positive (or zero): | (when a ≥ 0 ) |
Example:
... Or as soon as the exponent is odd : | (when n is odd ) |
Example:
... However when a is negative and the exponent is even we obtain this:
Did you watch that −3 ended up being +3 ?
| ... So we should do this: | (when a |
The |a| method the absolute value of a, in various other words any an unfavorable becomes a positive.
Example:
So the is something to be mindful of! Read an ext at index number of negative Numbers
Here it is in a little table:
nth root of a-to-the-mth-Power
What happens when the exponent and root are various values (m and also n)?
Well, us are permitted to readjust the order choose this:
n√am = (n√a )m
So this: nth source of (a come the strength m)becomes (nth source of a) come the power m
But over there is one even an ext powerful method ... Us can incorporate the exponent and also root to make a brand-new exponent, choose this: