## The nth Root

etc!

2 | √a × √a = a | The square root supplied two times in a multiplication provides the original value.You are watching: How to find nth root on scientific calculator | ||

3 | 3√a × 3√a × 3√a = a | The cube root provided three time in a multiplication gives the original value. | ||

n | n√a × n√a × ... × n√a = a(n the them) | The nth root provided n time in a multiplication offers the initial value. |

## The nth source Symbol

**This is the distinct symbol that way "nth root", that is the "radical"** price (used for square roots) v a little **n** to average **nth** root.

## Using it

We might use the nth root in a question like this:

Question: What is "n" in this equation?

n√625 = 5

Answer: i just occur to understand that **625 = 54** , for this reason the **4**th root of 625 must be 5:

4√625 = 5

## Why "Root" ... ?

When you view "root" think "I know the tree, but what is the root that created it? " Example: in |

## Properties

Now we know what an nth root is, let united state look at part properties:

### Multiplication and Division

We can "pull apart" multiplications under the root sign favor this:

n√ab = n√a × n√b **(Note: if n is also then a and b have to both be ≥ 0)**

This can help us leveling equations in algebra, and additionally make some calculations easier:

### Example:

3√128 = 3√64×2 = 3√64 × 3√2 = 43√2

for this reason the cube root of 128 simplifies come 4 times the cube source of 2.

It likewise works because that division:

n√a/b = n√a / n√b (a≥0 and b>0)Note the b can not be zero, as we can"t division by zero

### Addition and also Subtraction

**But we cannot** carry out that sort 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 same as **c = a + b** , right?

It is an easy trap to loss into, therefore beware.

It also method that, unfortunately, enhancements and subtractions can be hard to resolve when under a source sign.

### Exponents vs Roots

An exponent on one side of "=" have the right to be turned into a source on the various other side the "=":

If **an = b** climate **a = n√b**

Note: once n is even then b need to be ≥ 0

### nth source of a-to-the-nth-Power

When a value has actually an **exponent the n** and we take the **nth root** us **get the value earlier again** ...

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... As soon as a is | (when a ≥ 0 ) |

Example:

... Or when the | (when n is odd ) |

Example:

... But when **a is negative** and the **exponent is even** we get this:

Did you see that −3 came to be +3 ?

... Therefore we should do this: | (when a |

The |a| means the absolute worth of **a**, in other words any an unfavorable becomes a positive.

Example:

So that is something come be careful of! Read more at exponents of an unfavorable Numbers

Here the is in a small table:

n is odd n is even a ≥ 0 a

### nth root of a-to-the-mth-Power

What happens once the exponent and also root are various values (**m** and also **n**)?

Well, us are permitted to change the order favor this:

n√am = (n√a )m

So this: nth source of (a come the strength m)**becomes (nth root of a) come the strength m**

**But over there is an even an ext powerful method** ... Us can combine the exponent and also root to make a new exponent, like this: