# How to quickly calculate 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

### What is the square root?

The square root of **c** is the one **non-negative number**,

which multiplies by itself **c** results.

You also write $$ sqrt (c) $$ for the square root of c.

**Example:**

$$ sqrt (4) = 2 $$, since $$ 2 * 2 = 4 $$ **BUT**: $$ sqrt (4)! = -2 $$, although $$ (- 2) * (- 2) = 4 $$!

The root is always non-negative, so it cannot be $$ - 2 $$.

The pulling of the roots is also called **Square root**.

The number under the root is called **Radicand**.

square root

$$ uarr $$

$$ sqrt9 = 3 $$

$$ darr $$

Radicand

### Important connections

Squaring and rooting are **Reverse operations**.

You can do one process again through the other **undone** do.

### Get square roots from negative numbers?

You can only take square roots **non-negative** Draw numbers,

because the product of two equal numbers is always positive.

**Example:**

$$ sqrt (-4) $$ does not exist,

since $$ 2 * 2 = 4 $$ and $$ (- 2) * (- 2) = 4 $$

There is no such thing as a number that, when multiplied by itself, gives $$ - 4 $$

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### Extracting square roots from natural numbers

You can use roots from natural numbers **always pull**.

It is helpful to have the square numbers from $$ 1 ^ 2 $$ to $$ 25 ^ 2 $$ in mind.

The best thing to do is to memorize the square numbers. Then the tasks will fall to you too **without a calculator** light.

If you know that $$ 25 ^ 2 = 625 $$, you can easily take the square root of $$ 625 $$.

**Examples:**

$$ sqrt (25) = 5 $$ da $$ 5 * 5 = 25 $$

$$ sqrt (169) = 13 $$ da $$ 13 * 13 = 169 $$

$$ sqrt (0) = 0 $$ da $$ 0 * 0 = 0 $$ and $$ 0ge0 $$

### Extracting square roots from fractions

If you take square roots of fractions, you can

gradually **Numerator and denominator separate** consider.

The square numbers will also help you with fractions.

**Examples:**

$$ sqrt (25/36) = 5/6 $$ da $$ 5/6 * 5/6 = 25/36 $$

$$ sqrt (81/100) = 9/10 $$ da $$ 9/10 * 9/10 = 81/100 $$

$$ sqrt (9/441) = 3/21 = 1/7 $$ da $$ 3/21 * 3/21 = 9/441 $$

Finally, remember that you can shorten fractions.

### Get square roots from decimal fractions

If you want to get the root of a decimal fraction, think away the comma and remember the square numbers again.

**Examples:**

step | $$ sqrt (1.44) $$ | $$ sqrt (0.0576) $$ |
---|---|---|

Think away the comma and take root. | $$ sqrt (144) = 12 $$ | $$ sqrt (576) = 24 $$ |

Reason | $$12*12=144$$ | $$24*24=576$$ |

Insert decimal places. The result only has half as many decimal places like the radicand. | $$ sqrt (1.44) = 1.2 $$ | $$ sqrt (0.0576) = 0.24 $$ |

**BUT:** You can't just pull $$ sqrt (2.5) $$ because $$ 5 * 5 = 25 $$ and $$ 0.5 * 0.5 = 0.25 $$.

**Further examples:**

$$ sqrt (0.25) = 0.5 $$

$$ sqrt (6.25) = 2.5 $$

$$ sqrt (0.0001) = 0.01 $$

$$ sqrt (-0.09) $$ does not exist.

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### Square roots - now also double

Sometimes you come across tasks in which you suddenly see two root characters $$ sqrt (sqrt (m)) $$.

Then proceed gradually. You start with the **inner** Root. You pull the root again from the result. You can do that without a calculator.

**Example:**

$$ sqrt (sqrt (16)) = sqrt (4) = 2 $$

$$ sqrt (sqrt (81)) = sqrt (9) = 3 $$

### Powers under square roots

For example, if you were to calculate $$ sqrt (10 ^ 4) $$, consider the following:

$$ sqrt (10 ^ 4) = sqrt (10 * 10 * 10 * 10) $$

$$ = sqrt (10 ^ 2 * 10 ^ 2) $$

$$ = sqrt (10 ^ 2) * sqrt (10 ^ 2) $$

$$=10*10=10^2$$

You see: you halve the exponent and leave out the root sign. This is how you solve such tasks.

**Further examples:**

$$ sqrt (3 ^ 8) = sqrt (3 ^ 2 * 3 ^ 2 * 3 ^ 2 * 3 ^ 2) = 3 ^ 4 $$

$$ sqrt (10 ^ 12) = 10 ^ 6 $$

$$ sqrt (1 / (10 ^ 22)) = 1 / (10 ^ 11) $$

Form powers of two.

### Roots with the formula editor

This is how you enter roots in kapiert.de using the formula editor:

*kapiert.de*can do more:

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and tests - individual classwork trainer
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