I'm not sure if anyone saw my question yesterday, but here it is again. Even if you can only answer 1 or 2 of the questions, that is fine. I am so confused with the rules of exponents. I know them, but I have a hard time applying them because of all the steps. I was wondering if someone would be willing to answer these questions, because I am unsure if my answers are right.

x to the third power y to the negative two power/x to the negative two power times y to the fourth power=

2x to the neg. second power (3x to the fourth - 6x to the econd)=

(-2x to the third)(-3x to the neg second) both expressions to the neg second

(2x to the second y to the negative second) to the third power

(2x to the negative fifth y to the fourth) to the third

Thank you your help is genuinely appreciated.

x^3 y^-2 / [ x^-2 y^4]

3 - -2 = 5
-2 -4 = -6
so
x^5 y^-6

2 x^-2 ( 3 x^4 - 6 x^2 )
-2+4 = 2
-2 +2 = 0 remember anything^0 = 1
so
6 x^2 -12 x^0 = 6 x^2 -12

Sure, I'd be happy to help you with your questions on exponents! Let's go through them one by one.

1. x to the third power y to the negative two power / x to the negative two power times y to the fourth power:

To simplify this expression, we can apply the rules of exponents. When dividing terms with the same base, we subtract the exponents. So, for x terms, we have x^3 / x^(-2), which is equal to x^(3 - (-2)) or x^5. For y terms, we have y^(-2) / y^4, which is equal to y^(-2 - 4) or y^(-6). Finally, combining the x and y terms, we get x^5 * y^(-6).

2. 2x to the neg. second power (3x to the fourth - 6x to the second):

First, let's expand the expression inside the parentheses: 3x^4 - 6x^2. Now, we can apply the rule of exponents for multiplication. When raising a power to another power, we multiply the exponents. So, we have (2x^(-2))(3x^4 - 6x^2). Distribute the 2x^(-2) to both terms inside the parentheses: (2x^(-2))(3x^4) - (2x^(-2))(6x^2). Simplifying further, we get 6x^2 - 12x^(-2).

3. (-2x to the third)(-3x to the neg second) both expressions to the neg second:

To simplify this expression, we apply the rule of exponents for multiplication. When multiplying terms with the same base, we add the exponents. So, we have (-2x^3)(-3x^(-2))^(-2). Simplify further by distributing -2 to both terms inside the parentheses: (-2x^3)(-3^(-2)x^(-4)). Finally, combine the x terms by adding the exponents: -2 * -3^(-2) * x^(3 - 4), which further simplifies to 6/(-3^2x) or -6/(9x).

4. (2x to the second y to the negative second) to the third power:

To simplify this expression, we apply the rule of exponents for raising a power to another power. We multiply the exponents. So, we have (2x^2y^(-2))^3. Distribute the exponent 3 to both terms inside the parentheses: (2^3)(x^(2 * 3))(y^(-2 * 3)). Simplifying further, we get 8x^6y^(-6).

5. (2x to the negative fifth y to the fourth) to the third:

Similarly, to simplify this expression, we apply the rule of exponents for raising a power to another power. We multiply the exponents. So, we have (2x^(-5)y^4)^3. Distribute the exponent 3 to both terms inside the parentheses: (2^3)(x^(-5 * 3))(y^(4 * 3)). Simplifying further, we get 8x^(-15)y^12.

I hope this helps! Let me know if you have any further questions.