https://r2.erweima.ai/mathbot/image/1726690951747j2mbuwqm.jpg

Let's simplify each expression in the image.

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**a. \( \sqrt[30]{16x^8} \)**

We can rewrite the expression by breaking it into two parts:

\[
\sqrt[30]{16x^8} = \sqrt[30]{16} \times \sqrt[30]{x^8}
\]

- \( 16 = 2^4 \), so \( \sqrt[30]{16} = \left(2^4\right)^{1/30} = 2^{4/30} = 2^{2/15} \).
- \( \sqrt[30]{x^8} = x^{8/30} = x^{4/15} \).

Thus, the simplified expression is:

\[
2^{2/15} x^{4/15}
\]

---

**b. \( \sqrt{45} \times \sqrt{12} \times \sqrt{2} \)**

We can combine all the square roots under a single radical:

\[
\sqrt{45} \times \sqrt{12} \times \sqrt{2} = \sqrt{45 \times 12 \times 2} = \sqrt{1080}
\]

Now, we simplify \( \sqrt{1080} \):

\[
1080 = 2^3 \times 3^3 \times 5
\]

Taking the square root:

\[
\sqrt{1080} = \sqrt{(2^2 \times 3^2) \times 30} = 6\sqrt{30}
\]

Thus, the simplified expression is:

\[
6\sqrt{30}
\]

---

**c. \( \frac{c^3 \sqrt[3]{8b^4}}{10b^2c^2} \)**

First, simplify \( \sqrt[3]{8b^4} \):

\[
\sqrt[3]{8b^4} = \sqrt[3]{8} \times \sqrt[3]{b^4} = 2 \times b^{4/3}
\]

Now substitute into the expression:

\[
\frac{c^3 \times 2b^{4/3}}{10b^2c^2} = \frac{2c^3b^{4/3}}{10b^2c^2}
\]

Simplify by canceling common terms:

\[
= \frac{2cb^{4/3}}{10b^2} = \frac{c}{5b^{2 - 4/3}} = \frac{c}{5b^{2/3}}
\]

Thus, the simplified expression is:

\[
\frac{c}{5b^{2/3}}
\]

---

**d. \( \frac{2\sqrt{a^4}}{4a^2} \)**

Simplify \( \sqrt{a^4} \):

\[
\sqrt{a^4} = a^2
\]

Now substitute:

\[
\frac{2a^2}{4a^2} = \frac{2}{4} = \frac{1}{2}
\]

Thus, the simplified expression is:

\[
\frac{1}{2}
\]

---

If you have any questions or need further clarifications, feel free to ask! 

Here are five follow-up questions to deepen understanding:

1. How do you apply properties of exponents to simplify radicals with variables?
2. How do you simplify expressions with cube roots involving variables?
3. What are the steps to combine multiple square roots under a single radical?
4. How do you simplify expressions with fractions that contain both constants and variables?
5. How do you handle cube roots in the numerator when simplifying a fraction?

**Tip:** When simplifying radicals with exponents, always check if any part of the radical can be rewritten as a perfect power to make further simplifications easier.

Simplifica las siguientes expresiones: (a) √[30]{16x^8}, (b) √45 * √12 * √2, (c) c^3 √[3]{8b^4}/(10b^2c^2), (d) 2√a^4 / 4a^2

Learn how to simplify expressions involving radicals, exponents, and cube roots. This includes examples such as √[30]{16x^8}, √45 * √12 * √2, and more. Explore key algebraic concepts and step-by-step solutions for Grades 9-12.

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