Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
