Which is equivalent to sqrt(10) * 3/4 * ? X (root(10, 3)) ^ (4x) (root(10, 4)) ^ (3x) (root(10, 6)) ^ (4x) (root(10, 8)) ^ (3x)
The Correct answer and Explanation is :
The correct answer is: {1}{10}}
To find an expression equivalent to (\sqrt{10} \cdot \frac{3}{4} \cdot ?) in terms of the given roots of 10, we first need to understand the terms involved.
Definitions:
- Roots of 10:
- (\sqrt[3]{10} = 10^{1/3})
- (\sqrt[4]{10} = 10^{1/4})
- (\sqrt[6]{10} = 10^{1/6})
- (\sqrt[8]{10} = 10^{1/8})
- Square Root of 10:
- (\sqrt{10} = 10^{1/2})
Given Expression:
We are provided with the expression:
[
\sqrt{10} \cdot \frac{3}{4} \cdot ?
]
We want to find (?) such that:
[
\sqrt{10} \cdot \frac{3}{4} \cdot ? = \text{some combination of the roots}
]
Equating Powers of 10:
Let’s express the given terms with exponents:
- (\sqrt{10} = 10^{1/2})
- Let (?) be of the form (10^k) where (k) is to be determined.
Now, we can express the equation as:
[
10^{1/2} \cdot \frac{3}{4} \cdot 10^k = 10^{1/2 + k} \cdot \frac{3}{4}
]
To find the equivalent expression, we need to analyze each of the given roots and their exponents:
- ((\sqrt[3]{10})^{4x} = 10^{(1/3) \cdot 4x} = 10^{\frac{4x}{3}})
- ((\sqrt[4]{10})^{3x} = 10^{(1/4) \cdot 3x} = 10^{\frac{3x}{4}})
- ((\sqrt[6]{10})^{4x} = 10^{(1/6) \cdot 4x} = 10^{\frac{4x}{6}} = 10^{\frac{2x}{3}})
- ((\sqrt[8]{10})^{3x} = 10^{(1/8) \cdot 3x} = 10^{\frac{3x}{8}})
Finding the Right Expression:
To find (?), we need (1/2 + k) to equal one of the exponents from the roots. This leads us to:
[
\frac{3}{4} + k = 0 \implies k = -\frac{1}{4}
]
Thus,
[
? = 10^{-1/4} = \sqrt[4]{\frac{1}{10}}
]
Conclusion:
Therefore, the correct value of (?) that makes the expression equivalent is:
[
\sqrt[4]{\frac{1}{10}}.
]
This approach demonstrates how to manipulate the exponents and roots of 10 to find an equivalent expression, emphasizing the importance of understanding properties of exponents in algebra.