Answer :
Let's solve the given mathematical expressions step by step.
- For the expression [tex]\log_5 250 - \log_5 2 + \sqrt[4]{48}[/tex]:
Use the logarithmic property: [tex]\log_b a - \log_b c = \log_b \frac{a}{c}[/tex].
So, [tex]\log_5 250 - \log_5 2 = \log_5 \frac{250}{2} = \log_5 125[/tex].
Now, simplify [tex]\log_5 125[/tex].
Since $125 = 5^3[tex],[/tex]\log_5 125 = \log_5 (5^3) = 3[tex]because[/tex]\log_b (b^n) = n$.
Next, calculate [tex]\sqrt[4]{48}[/tex].
The fourth root of 48 can be simplified by finding the prime factorization of 48. Break it down:
$48 = 2^4 \times 3[tex]. Therefore,[/tex]\sqrt[4]{48} = \sqrt[4]{2^4 \times 3} = \sqrt[4]{2^4} \cdot \sqrt[4]{3} = 2 \cdot \sqrt[4]{3}$.
We assume [tex]\sqrt[3]{3}[/tex] is meant to indicate the cube root, but it seems not relevant to this expression.
Combine the results:
$3 + 2 \cdot \sqrt[4]{3}$ is an expression that might be simplified further depending on the values or context, but generally, that's the simplified form without numerical approximation of roots.
- For the expression [tex]\sqrt[4]{20000} : \sqrt[4]{2^7} \cdot \log_{0.1} 1000[/tex]:
Calculate [tex]\sqrt[4]{20000}[/tex].
Factor $20000$:
$20000 = 2^4 \times 5^4[tex]. So,[/tex]\sqrt[4]{20000} = \sqrt[4]{2^4 \times 5^4} = 2 \cdot 5 = 10$.
Calculate [tex]\sqrt[4]{2^7}[/tex].
[tex]\sqrt[4]{2^7} = 2^{7/4} = 2^{1.75}[/tex].
Calculate [tex]\log_{0.1} 1000[/tex].
[tex]\log_{0.1} 1000 = \frac{\log_{10} 1000}{\log_{10} 0.1}[/tex].
[tex]= \frac{3}{-1} = -3[/tex] (since [tex]\log_{10} 1000 = 3[/tex] and [tex]\log_{10} 0.1 = -1[/tex]).
Combine the results:
[tex]\frac{10}{2^{1.75}} \cdot (-3)[/tex]
However, without approximating $2^{1.75}$, the expression remains as above in more comprehensible terms without precise decimal simplification.
These calculations can be approached further by considering specific decimal values or context, but the essence is uncovering these components.