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:

 

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