## 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 **4**th 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 |

## 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 | (when a ≥ 0 ) |

Example:

... Or as soon as the | (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:

n is weird n is also a ≥ 0 a

### 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: