1. To simplify 49^(3/2), we can first convert the exponent into a fractional exponent. The exponent 3/2 means the square root of 49 cubed. The square root of 49 is 7, so the simplified expression becomes 7^3, which is equal to 7 * 7 * 7 = 343. Therefore, your answer is correct.
2. To simplify 5^x^25 * y^125, we can rewrite it as (5^x * y^25)^5. This means that both 5 and y are raised to the power of 25, while the exponent x is applied to 5 only. Therefore, the simplified expression is x^5 * y^25. So your answer is correct.
3. The given polynomial is 5x - 2x + 8x^2 - 7. When written in standard form, we rearrange the terms in descending order of their degree. Therefore, the leading coefficient is the coefficient of the term with the highest degree, which in this case is 8x^2. So your answer is incorrect; the correct leading coefficient is 8.
4. The polynomial 5x^2 + 9x + 1 has three terms and a degree of 2, making it a quadratic trinomial. So your answer is correct.
5. The area of the cage is given by the expression -2x^2 + 100w, where w is the width. To find the area when the width is 15 feet, we substitute w = 15 into the expression: -2x^2 + 100 * 15. Simplifying further, we get -2x^2 + 1500. Therefore, the area of the cage is 1500 square feet, not 1050 as you mentioned. So your answer is incorrect.
6. To add (5x^2 - 2x + 9) + (2x^2 - 4), we combine like terms. The terms with the same power of x can be added together. Thus, the sum becomes (5x^2 + 2x^2) + (-2x) + 9 + (-4). Combining like terms further, we get 7x^2 - 2x + 5. Therefore, your answer is incorrect; the correct sum is 7x^2 - 2x + 5.
7. To subtract (10a^2 - 6a) - (7a^2 - 8a), we distribute the negative sign to the terms inside the second parentheses. This gives us: 10a^2 - 6a - 7a^2 + 8a. Combining like terms, the result is 3a^2 + 2a. Therefore, your answer is incorrect; the correct difference is 3a^2 + 2a.
8. To multiply (-5rs^4)(3r^5s^2), we multiply the coefficients and add the exponents of the variables. Therefore, (-5 * 3)(r^1 * r^5)(s^4 * s^2) simplifies to -15r^6s^6. So your answer is correct.
9. To expand (3x - 4)^2, we square each term. Using the FOIL method, we get (3x)^2 - 2 * 3x * 4 + 4^2 = 9x^2 - 24x + 16. Therefore, your answer is incorrect; the correct expansion is 9x^2 - 24x + 16.
10. To multiply (2x + 4)(2x - 4), we use the distributive property. This gives us 2x * 2x + 2x * (-4) + 4 * 2x + 4 * (-4). Simplifying further, we get 4x^2 - 8x + 8x - 16, which simplifies to 4x^2 - 16. Therefore, your answer is incorrect; the correct product is 4x^2 - 16.