Sure! Let's go through a step-by-step process to find the value of the expression [tex]\(\sqrt[4]{6} \cdot \sqrt[3]{7} \cdot \sqrt{5}\)[/tex].
### Step 1: Calculate the fourth root of 6
The fourth root of a number [tex]\(a\)[/tex] can be written as [tex]\(a^{1/4}\)[/tex]. So, for [tex]\(6\)[/tex]:
[tex]\[
\sqrt[4]{6} = 6^{1/4} \approx 1.565
\][/tex]
### Step 2: Calculate the cube root of 7
The cube root of a number [tex]\(a\)[/tex] can be written as [tex]\(a^{1/3}\)[/tex]. So, for [tex]\(7\)[/tex]:
[tex]\[
\sqrt[3]{7} = 7^{1/3} \approx 1.913
\][/tex]
### Step 3: Calculate the square root of 5
The square root of a number [tex]\(a\)[/tex] can be written as [tex]\(a^{1/2}\)[/tex] or [tex]\(a^{0.5}\)[/tex]. So, for [tex]\(5\)[/tex]:
[tex]\[
\sqrt{5} = 5^{0.5} \approx 2.236
\][/tex]
### Step 4: Multiply the results from Steps 1, 2, and 3
We now multiply the three calculated roots:
[tex]\[
(6^{1/4}) \cdot (7^{1/3}) \cdot (5^{0.5}) \approx 1.565 \cdot 1.913 \cdot 2.236
\][/tex]
Calculating the product:
[tex]\[
1.565 \cdot 1.913 \cdot 2.236 \approx 6.695
\][/tex]
Thus, the value of [tex]\(\sqrt[4]{6} \cdot \sqrt[3]{7} \cdot \sqrt{5}\)[/tex] approximately equals 6.695.
