Let's break down and solve each of the equations step by step.

1. $x^5 \times x^3$:

   • When multiplying powers with the same base, add the exponents: $x^{5+3} = x^8$

2. $- (7x)$:

   • This simply means the negative of $7x$, so: $-7x$

3. $(a + b^3)^2$:

   • Apply the square to both terms inside the parentheses: $(a + b^3)(a + b^3) = a^2 + 2ab^3 + b^6$

4. $\frac{6x^8}{2x^4}$:

   • Simplify by dividing the coefficients (6/2) and subtracting the exponents of $x$: $\frac{6}{2} \times x^{8-4} = 3x^4$

5. $(3x^2y^{-3})^{-2}$:

   • Apply the power of $-2$ to both the coefficient and the exponents inside: $(3^{-2}) \times (x^{2 \times -2}) \times (y^{-3 \times -2}) = \frac{1}{9} \times x^{-4} \times y^6 = \frac{y^6}{9x^4}$

6. $x^9 \times x^{-7}$:

   • When multiplying powers with the same base, add the exponents: $x^{9 + (-7)} = x^2$

7. $(4x^2y^{-3})^2$:

   • Apply the square to both the coefficient and the exponents: $(4^2) \times (x^{2 \times 2}) \times (y^{-3 \times 2}) = 16x^4y^{-6} = \frac{16x^4}{y^6}$

8. $\frac{1}{2^4}$:

   • Simplify the power of 2: $\frac{1}{2^4} = \frac{1}{16}$

9. $\frac{-7a^2b^3c^0}{3a^3b^2c^7}$:

   • Simplify by subtracting the exponents of the like terms: $\frac{-7}{3} \times a^{2-3} \times b^{3-2} \times c^{0-7} = \frac{-7b}{3a^1c^7} = \frac{-7b}{3ac^7}$

10. $(2c \times d^2)^3 \times (c \times d^4)^2$:

    • First, apply the exponents separately to each term:
    • $(2c \times d^2)^3 = 2^3 \times c^3 \times d^6 = 8c^3d^6$
    • $(c \times d^4)^2 = c^2 \times d^8$
    • Multiply the two results: $8c^3d^6 \times c^2d^8 = 8c^{3+2}d^{6+8} = 8c^5d^{14}$

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Do you want more details or have any questions?

Here are some related questions:

1. How do you simplify expressions with multiple variables?
2. What is the rule for distributing exponents over a product?
3. How do you simplify expressions with negative exponents?
4. What is the power of a power rule in exponents?
5. How do you divide expressions with the same base?
6. How do you multiply polynomials using the distributive property?
7. What is the difference between a polynomial and a monomial?
8. How do you deal with zero exponents?

Tip: When working with exponents, always be mindful of negative exponents and how they affect the overall expression.