Can we subtract a variable from zero

Summarize square roots with variables

Just like you group square roots with numbers, so can you Roots with variables sum up.

Examples of root terms with variables:

$$ sqrt (z * z ^ 3) $$ $$ sqrt (from ^ 2) $$ $$ sqrt (a / (from ^ 2)) $$

In the following you will learn again the laws of roots for Products and Quotient and you can look at examples with variables.

In memory of:

You can't just add or subtract roots.

Correct: $$ sqrt (25) -sqrt (16) = 5-4 = 1 $$

Adhere to the definition range of variables

For tasks with variables, you first look Which numbers you can use for the variables.

You cannot take roots from negative numbers and the root can never be negative.

Case 1:

As a rule, the variables are

Example: $$ sqrt (z * z ^ 2) $$ for

Case 2:

Sometimes you can substitute for the variable.

Example: $$ sqrt (z * z ^ 3) $$ for

Multiply square roots

Case 1: Variable $$ ge0 $$

We'll limit ourselves to that for now non-negative Radicands.
The following law of roots applies to products of square roots:

$$ sqrt (a) * sqrt (b) = sqrt (a * b) $$ with

You multiply two square roots by taking the Multiplying radicands and then from the product
take root.

Example: $$ sqrt (z) * sqrt (z ^ 3) = sqrt (z * z ^ 3) = sqrt (z ^ 4) = z ^ 2 $$

Proof:

  • First of all, $$ sqrt (a) * sqrt (b) $$ is not negative because $$ sqrt (a) $$ and $$ sqrt (b) $$ are not negative.

  • $$ (sqrt (a) * sqrt (b)) ^ 2 $$
    $$ = (sqrt (a) * sqrt (b)) * (sqrt (a) * sqrt (b)) $$
    $$ = sqrt (a) * sqrt (a) * sqrt (b) * sqrt (b) $$
    $$ = a * b $$

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Divide square roots

Case 1: Variable $$ ge0 $$

First consider non-negative Radicands.

The following law of roots applies to quotients of square roots:

$$ sqrt (a) / sqrt (b) = sqrt (a / b) $$ with and

You divide two square roots by taking the
Dividing radicands and then from the quotient
take root.

$$ sqrt (a): sqrt (from ^ 2) = sqrt (a) / sqrt (from ^ 2) = sqrt (a / (from ^ 2)) $$

$$ stackrel (truncate) = sqrt (1 / b ^ 2) = sqrt (1) / sqrt (b ^ 2) = 1 / b $$ with

Proof:

  • First, $$ sqrt (a): sqrt (b) $$ is non-negative, since $$ sqrt (a) $$ and $$ sqrt (b) $$ are non-negative.
  • $$ (sqrt (a): sqrt (b)) ^ 2 $$
    $$ = (sqrt (a) / sqrt (b)) ^ 2 $$
    $$ = (sqrt (a) / sqrt (b)) * (sqrt (a) / sqrt (b)) $$
    $$ = a / b $$

Reshape root terms

Case 1: Variable $$ ge0 $$

So you bring one factor under the root:

You can use variables just like numbers by squaring write under a root. Then you apply the laws of the roots.

Example: $$ c * sqrt (7) = sqrt (c ^ 2) * sqrt (7) = sqrt (7 * c ^ 2) $$ with

Reshape root terms

Case 1: Variable $$ ge0 $$

That's how it is done partial pulling of roots:

Find the square number in the radicand. You can only "remove" variables from the root if they have an even exponent.

Examples:

a) $$ sqrt (a / 49) = sqrt (a) / sqrt (49) = sqrt (a) / 7 $$

b) $$ sqrt ((a ^ 2b ^ 3) / (18z ^ 2)) = sqrt (a ^ 2b ^ 3) / sqrt (18z ^ 2) = (a * sqrt (b ^ 3)) / (z * sqrt (9 * 2)) = (asqrt (b ^ 3)) / (3zsqrt (2)) = a / (3z) * sqrt (b ^ 3/2) $$

and

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Special cases

Case 2: variable $$ inRR $$

A root is always non-negative. A negative number can never come out. You can do that with Amount bars express.

Example: $$ sqrt ((- 4) ^ 2) = | -4 | = 4 $$

Attention, this is wrong:

In general: $$ sqrt (a ^ 2) = | a | $$


Examples: Partially pull the root.

a) $$ sqrt (a ^ 2 * b) = sqrt (a ^ 2) * sqrt (b) = | a | * sqrt (b) $$ with and

b) $$ sqrt ((a ^ 2b ^ 3) / (18z ^ 2)) = sqrt (a ^ 2b ^ 3) / sqrt (18z ^ 2) = (| a | * sqrt (b ^ 3)) / (| z | * sqrt (9 * 2)) = (| a | sqrt (b ^ 3)) / (3 | z | sqrt (2)) $$$$ = | a | / (3 | z |) * sqrt (b ^ 3/2) $$
with and

The amount

... is a non-negative number that indicates the distance to zero for any number.

Example: $$ | 3 | = 3 $$ and $$ | -3 | = 3 $$

This is how you reshape root terms

  1. Look in the task to see which numbers you have
    for the variable.

  2. Case 1: Variable $$ ge0 $$
    Apply the laws of the roots as learned.

    Case 2: Variable $$ in RR $$
    Calculate with the amount bars.
    $$ sqrt (a ^ 2) = | a | $$

Roots with the formula editor

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

 

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