To simplify the given expression:

$2^{\log_2 5} - 3 \log_5 \sqrt[3]{5}$

Let's break it down step by step.

1. Simplify $2^{\log_2 5}$:

This part involves a logarithm and an exponent with the same base. One of the properties of logarithms is:

$a^{\log_a x} = x$

Using this property for $2^{\log_2 5}$:

$2^{\log_2 5} = 5$

2. Simplify $3 \log_5 \sqrt[3]{5}$:

First, we rewrite $\sqrt[3]{5}$ as an exponent:

$\sqrt[3]{5} = 5^{1/3}$

Next, apply the logarithmic property:

$\log_b(x^n) = n \log_b x$

This gives:

$\log_5 \sqrt[3]{5} = \log_5(5^{1/3}) = \frac{1}{3} \log_5 5$

Since $\log_5 5 = 1$, we have:

$\log_5 \sqrt[3]{5} = \frac{1}{3}$

Now, multiply by 3:

$3 \log_5 \sqrt[3]{5} = 3 \times \frac{1}{3} = 1$

3. Final Expression:

Now that both parts are simplified, we can subtract them:

$5 - 1 = 4$

Thus, the simplified expression is:

$\boxed{4}$

Would you like further details on this?

Here are 5 related questions for practice:

1. Simplify $3^{\log_3 7} - 2 \log_7 \sqrt[4]{7}$.
2. What is $5^{\log_5 x}$ for any positive $x$?
3. Simplify $4 \log_2 \sqrt{8}$.
4. Use properties of logarithms to simplify $\log_3 (9 \times 27)$.
5. Solve for $x$: $2^{\log_2 (3x)} = 6$.

Tip: Remember that logarithmic and exponential functions are inverses of each other!