To analyze and solve this, I'll simplify each expression involving radicals. Here is the breakdown:

1. Expression 1: $\sqrt[4]{a^7b}$

The $n$-th root can be expressed as fractional exponents:

$\sqrt[4]{a^7b} = a^{\frac{7}{4}}b^{\frac{1}{4}}$

2. Expression 2: $\sqrt[5]{a^5b^6c^{12}}$

Each term inside the root is raised to the fractional exponent $\frac{1}{5}$:

$\sqrt[5]{a^5b^6c^{12}} = a^{\frac{5}{5}}b^{\frac{6}{5}}c^{\frac{12}{5}}$

Simplify $a^{\frac{5}{5}}$:

$a^{\frac{5}{5}} = a$

Thus:

$\sqrt[5]{a^5b^6c^{12}} = ab^{\frac{6}{5}}c^{\frac{12}{5}}$

3. Expression 3: $\sqrt[3]{a^7b^{15}c}$

Apply fractional exponents $\frac{1}{3}$:

$\sqrt[3]{a^7b^{15}c} = a^{\frac{7}{3}}b^{\frac{15}{3}}c^{\frac{1}{3}}$

Simplify $b^{\frac{15}{3}}$:

$b^{\frac{15}{3}} = b^5$

Thus:

$\sqrt[3]{a^7b^{15}c} = a^{\frac{7}{3}}b^5c^{\frac{1}{3}}$

4. Expression 4: $\sqrt[4]{4a^4b}$

Simplify each term:

• $4$ inside the $4$-th root becomes $4^{\frac{1}{4}}$.
• $a^4$ becomes $a^{\frac{4}{4}}$, which simplifies to $a^1$.
• $b$ becomes $b^{\frac{1}{4}}$.

So:

$\sqrt[4]{4a^4b} = 4^{\frac{1}{4}}a^1b^{\frac{1}{4}}$

Final Results:

1. $\sqrt[4]{a^7b} = a^{\frac{7}{4}}b^{\frac{1}{4}}$
2. $\sqrt[5]{a^5b^6c^{12}} = ab^{\frac{6}{5}}c^{\frac{12}{5}}$
3. $\sqrt[3]{a^7b^{15}c} = a^{\frac{7}{3}}b^5c^{\frac{1}{3}}$
4. $\sqrt[4]{4a^4b} = 4^{\frac{1}{4}}ab^{\frac{1}{4}}$

Would you like me to clarify any part or provide further steps?

Related Questions:

1. How do you calculate the fractional exponents for roots?
2. What happens when a base is raised to a fractional power?
3. How do we simplify terms with similar bases and fractional exponents?
4. What is the significance of simplifying radicals in mathematics?
5. How do these expressions relate to real-world applications like physics or engineering?

Tip:

When simplifying expressions with radicals, converting them to fractional exponents helps with understanding and combining terms.