Let's go through the given problems one by one and simplify them.

1. $4^3 + 4^7$

Using exponentiation:

• $4^3 = 64$
• $4^7 = 16,384$

Now add them together: $4^3 + 4^7 = 64 + 16,384 = 16,448$

2. $\frac{5^8}{5^5}$

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$: $\frac{5^8}{5^5} = 5^{8-5} = 5^3 = 125$

3. $(4^6)^3$

Using the power of a power rule $(a^m)^n = a^{m \cdot n}$: $(4^6)^3 = 4^{6 \cdot 3} = 4^{18}$ Now calculate $4^{18}$: $4^{18} = 68,719,476,736$

4. $(3c^5 \times 6^2 b^3 a^6)^5$

Apply the power to each term: $(3c^5 \times 6^2 b^3 a^6)^5 = 3^5 c^{25} \times 6^{10} b^{15} a^{30}$ Now calculate the numerical part: $3^5 = 243, \quad 6^{10} = 60,466,176$ So the final answer is: $243 \times 60,466,176 \times c^{25} b^{15} a^{30}$

5. $\sqrt{4} + \sqrt{4}$

Since $\sqrt{4} = 2$: $\sqrt{4} + \sqrt{4} = 2 + 2 = 4$

6. $2\sqrt{7} + 6\sqrt{7}$

Combine like terms: $2\sqrt{7} + 6\sqrt{7} = (2+6)\sqrt{7} = 8\sqrt{7}$

7. $9\sqrt{6} - 7\sqrt{6}$

Combine like terms: $9\sqrt{6} - 7\sqrt{6} = (9 - 7)\sqrt{6} = 2\sqrt{6}$

8. $\sqrt{4} - \sqrt{8}$

Since $\sqrt{4} = 2$ and $\sqrt{8} = 2\sqrt{2}$: $\sqrt{4} - \sqrt{8} = 2 - 2\sqrt{2}$ So the simplified form is: $2 - 2\sqrt{2}$

9. $7\sqrt{3} + 3\sqrt{4}$

Since $\sqrt{4} = 2$: $7\sqrt{3} + 3\sqrt{4} = 7\sqrt{3} + 3 \times 2 = 7\sqrt{3} + 6$

10. $\sqrt{7} \times \sqrt{7}$

Using the property $\sqrt{a} \times \sqrt{a} = a$: $\sqrt{7} \times \sqrt{7} = 7$

11. $\sqrt{3} \times \sqrt{7}$

Using the property $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$: $\sqrt{3} \times \sqrt{7} = \sqrt{3 \times 7} = \sqrt{21}$

Let me know if you need detailed steps for any of these! Here are 5 relative questions you might explore next:

1. How do exponent rules apply to multiplication and division of powers?
2. What are the different ways to simplify square roots involving multiplication?
3. How can we simplify expressions with multiple radicals?
4. What is the general formula for powers of powers in algebraic terms?
5. What are other properties of radicals that can simplify complex expressions?

Tip:

When working with radicals, always try to factor out perfect squares or simplify the radical before performing addition, subtraction, or multiplication.