Sure, let's go through the problem step-by-step to determine the value of [tex]\( a \times b \times c \)[/tex] given the equations:
1. [tex]\(\sqrt[a]{2} = 3\)[/tex]
2. [tex]\(\sqrt[b]{3} = 5\)[/tex]
3. [tex]\(\sqrt[c]{5} = 8\)[/tex]
### Step 1: Rewrite Each Root Expression
The expression [tex]\(\sqrt[a]{2} = 3\)[/tex] means that [tex]\( 2^{1/a} = 3 \)[/tex]. Taking the natural logarithm of both sides, we have:
[tex]\[ \log (2^{1/a}) = \log (3) \][/tex]
By using the logarithm properties, we get:
[tex]\[ \frac{\log (2)}{a} = \log (3) \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{\log (2)}{\log (3)} \][/tex]
### Step 2: Solve for [tex]\(b\)[/tex]
Similarly, for [tex]\(\sqrt[b]{3} = 5\)[/tex]:
[tex]\[ 3^{1/b} = 5 \][/tex]
Taking the natural logarithm of both sides:
[tex]\[ \log (3^{1/b}) = \log (5) \][/tex]
Using the logarithm properties:
[tex]\[ \frac{\log (3)}{b} = \log (5) \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = \frac{\log (3)}{\log (5)} \][/tex]
### Step 3: Solve for [tex]\(c\)[/tex]
For [tex]\(\sqrt[c]{5} = 8\)[/tex]:
[tex]\[ 5^{1/c} = 8 \][/tex]
Taking the natural logarithm of both sides:
[tex]\[ \log (5^{1/c}) = \log (8) \][/tex]
Using the logarithm properties:
[tex]\[ \frac{\log (5)}{c} = \log (8) \][/tex]
Solving for [tex]\(c\)[/tex]:
[tex]\[ c = \frac{\log (5)}{\log (8)} \][/tex]
### Step 4: Calculate [tex]\( a \times b \times c \)[/tex]
We now have expressions for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ a = \frac{\log (2)}{\log (3)} \][/tex]
[tex]\[ b = \frac{\log (3)}{\log (5)} \][/tex]
[tex]\[ c = \frac{\log (5)}{\log (8)} \][/tex]
Let's find the product [tex]\( a \times b \times c \)[/tex]:
[tex]\[ a \times b \times c = \left( \frac{\log (2)}{\log (3)} \right) \times \left( \frac{\log (3)}{\log (5)} \right) \times \left( \frac{\log (5)}{\log (8)} \right) \][/tex]
Notice that the logarithms cancel out sequentially:
[tex]\[ a \times b \times c = \frac{\log (2)}{\log (8)} \][/tex]
Since [tex]\(\log (8) = \log (2^3) = 3 \log (2)\)[/tex]:
[tex]\[ a \times b \times c = \frac{\log (2)}{3 \log (2)} = \frac{1}{3} \][/tex]
Therefore, the value of [tex]\( a \times b \times c \)[/tex] is:
[tex]\[ \boxed{\frac{1}{3}} \][/tex]
