1. x ^ 5 * x ^ 3 =

2.-(7x) =

3. (a+b³)2 =

4. (6x ^ 8)/(2x ^ 4) =

5. (3x²y-³) -2 =

6. x ^ 9 * x ^ - 7 =

7. (4x²y-3)2 =

8. 1/(2 ^ 4) =

9. (- 7a ^ 2 * b ^ 3 * c ^ 0)/(3a ^ 3 * b ^ 2 * c ^ 7) =

10. (2c * d ^ 2) ^ 3 * (c * d ^ 4) ^ 2 =

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.

Explore how to solve exponential and polynomial expressions with detailed step-by-step solutions. This includes understanding exponents, polynomials, and rules for negative and fractional exponents. Suitable for high school students.

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