Math  /  Algebra

QuestionSimplify. Assume that all variables represent positive numbers. z236x2z6x\frac{\sqrt{\frac{z^{2}}{36 x^{2}}}}{\frac{|z|}{6|x|}}
Suggested tutorial: Learn It: Simplify a radical expression.

Studdy Solution

STEP 1

What is this asking? We're taking a square root of a fraction with variables and then dividing it by another fraction with the same variables, and we need to simplify it as much as possible! Watch out! Remember those absolute value signs?
They're important, especially since we're told the variables are positive!

STEP 2

1. Simplify the numerator
2. Simplify the denominator
3. Divide and conquer

STEP 3

Alright, let's **tackle that square root** in the numerator!
Remember, the square root of a fraction is the same as the square root of the numerator divided by the square root of the denominator.
So, we have: z236x2=z236x2 \sqrt{\frac{z^{2}}{36 x^{2}}} = \frac{\sqrt{z^{2}}}{\sqrt{36 x^{2}}}

STEP 4

Now, we can **simplify** those square roots.
The square root of z2z^2 is z|z|, and the square root of 36x236x^2 is 6x6|x|.
So, our numerator becomes: z236x2=z6x \frac{\sqrt{z^{2}}}{\sqrt{36 x^{2}}} = \frac{|z|}{6|x|}

STEP 5

We're told that all variables represent **positive numbers**.
This means z|z| is just zz and x|x| is just xx.
So, our denominator becomes: z6x=z6x \frac{|z|}{6|x|} = \frac{z}{6x}

STEP 6

Now, we're **dividing** the simplified numerator by the simplified denominator: z6xz6x \frac{\frac{|z|}{6|x|}}{\frac{|z|}{6|x|}}

STEP 7

Let's **substitute** in our simplified versions, remembering that z=z|z| = z and x=x|x| = x since xx and zz are positive: z6xz6x \frac{\frac{z}{6x}}{\frac{z}{6x}}

STEP 8

Remember, dividing by a fraction is the same as **multiplying by its reciprocal**.
The reciprocal of z6x\frac{z}{6x} is 6xz\frac{6x}{z}.
So, we have: z6x6xz \frac{z}{6x} \cdot \frac{6x}{z}

STEP 9

Now we **multiply** across: z6x6xz=6xz6xz \frac{z}{6x} \cdot \frac{6x}{z} = \frac{6xz}{6xz} Since we're told that the variables are positive, we know that neither xx nor zz is zero.
We can **divide** both the numerator and the denominator by 6xz6xz to get: 6xz6xz=1 \frac{6xz}{6xz} = 1

STEP 10

Our **final simplified answer** is 11.

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