# How to calculate a square root

### What is the square root?

The square root of c is the one non-negative number,
which multiplied 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 \$\$

### 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 single number that, when multiplied by itself, yields \$\$ - 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.

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. From the result you pull the root again. 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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