# 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

- Look in the task to see which numbers you have

for the variable. - 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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