Sure, let's simplify the given mathematical expression step-by-step.
We start with the expression:
[tex]\[ \sqrt{z} \cdot \sqrt{30 z^2} \cdot \sqrt{35 z^3} \][/tex]
### Step 1: Simplify each square root term
First, let's recall that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. So, we can rewrite each term under a single square root:
[tex]\[ \sqrt{30 z^2} = \sqrt{30} \cdot \sqrt{z^2} \][/tex]
[tex]\[ \sqrt{35 z^3} = \sqrt{35} \cdot \sqrt{z^3} \][/tex]
### Step 2: Simplify the simplified square roots
Recall that [tex]\(\sqrt{z^n} = z^{n/2}\)[/tex]. Let’s simplify:
[tex]\[ \sqrt{z^2} = z \][/tex]
[tex]\[ \sqrt{z^3} = z^{3/2} \][/tex]
So we have:
[tex]\[ \sqrt{30 z^2} = \sqrt{30} \cdot z \][/tex]
[tex]\[ \sqrt{35 z^3} = \sqrt{35} \cdot z^{3/2} \][/tex]
### Step 3: Substitute back and combine
Substitute these back into the original expression:
[tex]\[ \sqrt{z} \cdot (\sqrt{30} \cdot z) \cdot (\sqrt{35} \cdot z^{3/2}) \][/tex]
### Step 4: Group all the [tex]\(\sqrt{}\)[/tex] terms and all the [tex]\(z\)[/tex] terms together
Group the [tex]\(\sqrt{}\)[/tex] terms:
[tex]\[ (\sqrt{z}) \cdot (\sqrt{30}) \cdot (\sqrt{35}) \cdot z \cdot z^{3/2} \][/tex]
To combine the [tex]\(\sqrt{}\)[/tex] terms:
[tex]\[ \sqrt{z \cdot 30 \cdot 35} \][/tex]
Calculate the product inside the square root:
[tex]\[ 30 \cdot 35 = 1050 \][/tex]
So we have:
[tex]\[ \sqrt{z \cdot 1050} \][/tex]
### Step 5: Combine the [tex]\(z\)[/tex] terms
Recall the exponents rule [tex]\(z^m \cdot z^n = z^{m+n}\)[/tex]:
[tex]\[ z \cdot z^{3/2} = z^{2/2 + 3/2} = z^{5/2} \][/tex]
### Step 6: Multiply all parts together
Combining the simplified parts, we get:
[tex]\[ \sqrt{1050} \cdot z^{5/2} \][/tex]
Note the values were calculated earlier:
[tex]\[ \sqrt{1050} \approx \sqrt{25 \cdot 42}=5\sqrt{42} \][/tex]
Thus, the simplified expression is:
[tex]\[ 5 \sqrt{42} \cdot z^{5/2} = 5 \sqrt{42} \cdot z^{3} \][/tex]
### Final Answer
The simplified form of the expression:
[tex]\[ \sqrt{z} \cdot \sqrt{30 z^2} \cdot \sqrt{35 z^3} \][/tex]
is:
[tex]\[ 5 \sqrt{42} \cdot z^{3} \][/tex]
