# What is the cube root of 3

### Reminder: the square root

You already know the square root. It is the "inversion" of "to the power of 2".**$$ sqrt121 = 11 $$**, because $$ 11 ^ 2 = 11 cdot 11 = 121 $$

The square root of $$ x $$ is the non-negative number which, when multiplied by itself, results in $$ x $$.

Your calculator can calculate roots. Still, it will help you if you have the square numbers in your head.

### What is the 3rd root?

You can not only reverse “to the power of 2”, but also “to the power of 3”! For this you need the 3rd root, or "cube root".

**$$ root 3 (8) = 2 $$**, because $$ 2 ^ 3 = 2 * 2 * 2 = 8 $$

3rd root

$$ uarr $$

$$ root 3 (8) = 2 $$

$$ darr $$

Radicand

$$ root 3 (a) = b $$ $$ rarr $$ The 3rd root is the non-negative number b, which results in the number a as the third power (b³).

$$ a $$ is a real, non-negative number: $$ a in RR $$ and $$ a ge 0 $$. Then $$ b in RR $$ and $$ b ge 0 $$ also holds

Taking the 3rd root is inverting the 3rd power.

The small 3 at the root sign means that you are taking the 3rd root.

### Geometric

### square

You calculate the area of a square with $$ A = a ^ 2 $$. Here, $$ a $$ is the length of the side.

So the opposite applies: $$ sqrtA = a $$

The root of the area $$ A = 9 $$ of the square is the side length $$ a = 3 $$.

$$ sqrt 9 = 3 $$, because $$ 3 ^ 2 = 9 $$.

### cube

How do you get the side length of a cube?

You calculate the volume $$ V $$ of a cube with $$ V = a ^ 3 $$. So $$ root (3) V = a $$.

The 3rd root of the volume $$ V = 8 $$ of the cube is the side length $$ 2 $$.

$$ root 3 (8) = 2 $$, because $$ 2 ^ 3 = 8 $$

The word "cube" comes from "cube". That means dice.

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### Some examples

$$ root 3 (1) = 1 $$, because $$ 1 ^ 3 = 1 $$

$$ root 3 (8) = 2 $$, because $$ 2 ^ 3 = 8 $$

$$ root 3 (27) = 3 $$, because $$ 3 ^ 3 = 27 $$

$$ root 3 (64) = 4 $$, because $$ 4 ^ 3 = 64 $$

$$ root 3 (125) = 5 $$, because $$ 5 ^ 3 = 125 $$

$$ root 3 (1 000) = 10 $$, because $$ 10 ^ 3 = 1000 $$

### Now also with a comma

Decimal numbers also have cube roots:

$$ root 3 (3.375) = 1.5 $$

$$ root 3 (0.125) = 0.5 $$

$$ root 3 (0.001) = 0.1 $$

$$ root 3 (15,625) = 2.5 $$

### And fractions

$$ root 3 (8/27) = 2/3 $$, because $$ (2/3) ^ 3 = 2/3 * 2/3 * 2/3 = 8/27 $$

$$ root 3 (1/125) = 1/5 $$, because $$ (1/5) ^ 3 = 1/5 * 1/5 * 1/5 = 1/125 $$

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### Irrational?

You have now seen a bunch of 3. roots that are natural numbers (64) or decimals (0.5) or fractions. Most of the 3rd roots are, however **irrational**, that is, non-terminating, non-periodic decimal numbers.

The calculator will help you with the calculation. Find the key for the 3rd root and type:

$$ root 3 (x) $$ $$ 15 $$

or

$$ 15 $$$$ root 3 (x) $$

and the calculator will give you $$ 2.4662120743 ... $$. The number of decimal places can vary, depending on how much space there is on your display.

Usually you should round to 2 decimal places:

$$ root 3 (15) approx 2.47 $$

You already know irrational numbers from the square roots. $$ sqrt2 $$ or $$ sqrt3 $$ are irrational numbers.

### Letter salad

You already guessed it: What works with numbers, also works with variables. :-) Variables just have to state which numbers you can use.

**Examples:**

$$ root 3 (x ^ 3) = x $$ - with $$ x ge0 $$

$$ root 3 (x ^ 6) = x ^ 2 $$, because $$ (x ^ 2) ^ 3 = x ^ 6 $$ - with $$ x ge0 $$

$$ root 3 (1 / y ^ 6) = 1 / y ^ 2 $$, because $$ (1 / y ^ 2) ^ 3 = 1 ^ 3 / ((y ^ 2) ^ 3) = 1 / y ^ 6 $$ - with $$ y ge0 $$

### Interval nesting

With the interval nesting you can find the 3rd root **approximately** calculate without using the root key of your calculator.

**Example:** $$ root 3 (52) $$

Note: You calculate the calculation steps marked in blue with the calculator.

### Step 1: Find the first interval

Between which natural numbers is $$ root 3 (52) $$?

- Try the cube numbers $$ 1 ^ 3 $$, $$ 2 ^ 3 $$, $$ 3 ^ 3 $$, $$ 4 ^ 3, ... $$.
- $$ 3 ^ 3 = 27 le 52 le 4 ^ 3 = 64 $$ applies. So $$ root 3 (52) $$ lies between $$ 3 $$ and $$ 4 $$.

### Step 2: Box the interval further

- Add a decimal place.
- Use the calculator to find out between which of the numbers $$ (3,1) ^ 3, (3,2) ^ 3, (3,3) ^ 3,…, (3,9) ^ 3 $$ the number $$ 52 $$ lies.
- $$ 3.7leroot 3 (52) le3.8 $$, because $$ (3.7) ^ 3 = 50.65 $$$$ le52le $$$$ (3.8) ^ 3 = 54.87 $$

### 3rd step: Two decimal places

- Use the calculator to calculate between which of the numbers $$ (3.71) ^ 3, (3.72) ^ 3, (3.73) ^ 3,…, (3.79) ^ 3 $$ the number $$ 52 $$ lies.
- $$ 3.73leroot 3 (52) le3.74 $$, because $$ (3.73) ^ 3 = 51.9 $$$$ le52le $$$$ (3.74) ^ 3 = 52.31 $$

### 3rd step: three decimal places

- Use the calculator to find out between which of the numbers $$ (3.731) ^ 3, (3.732) ^ 3, (3.733) ^ 3,…, (3.739) ^ 3 $$ the number $$ 52 $$ lies.
- $$ 3.732leroot 3 (52) le3.733 $$, because $$ (3.732) ^ 3 = 51.98 $$$$ le52le $$$$ (3.733) ^ 3 = 52.02 $$

With each step you narrow down $$ root 3 (52) $$ more precisely. Since $$ root 3 (52) $$ is irrational, you never get the exact value.

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