application of the laws of exponent

show different ways of simplifying each of the following expression. answer the question that follow.
1. 2^5 * 2^4
2. x^4 * x^7
3. (3^2)^3
4. (m^4)^5
5. (5^3 * 2^2)^2
6. (a^4 * b^2)^5
7. (3x^5)^3
8. 7^4/7^2
9. x^5/x^8
10. 12m^9/4m^5
11. (3/4)^2
12. (x/y)^5
13. (2x^5/5y^4)^3
14. (12m^4/6)^3
15. (3/9a^3)^4
B.
1. how did you simplify each expression above?
2. what do you hink would make it easy, to simplify the given expression?why?

When multiplying terms, you add the exponents. when dividing terms, you subtract exponents. With exponents of exponents, you multiply. For example,

3. (3^2)^3 = 3^2 * 3^2 * 3^2 = 3^6

To simplify each of the given expressions, we can apply the laws of exponents:

1. 2^5 * 2^4:
To simplify, we add the exponents since the bases (2) are the same.
2^5 * 2^4 = 2^(5+4) = 2^9

2. x^4 * x^7:
Again, we add the exponents since the bases (x) are the same.
x^4 * x^7 = x^(4+7) = x^11

3. (3^2)^3:
To simplify, we multiply the exponents inside and outside the parentheses.
(3^2)^3 = 3^(2*3) = 3^6

4. (m^4)^5:
We multiply the exponents.
(m^4)^5 = m^(4*5) = m^20

5. (5^3 * 2^2)^2:
We apply the exponent to both the 5 and 2 separately within the parentheses.
(5^3 * 2^2)^2 = (5^(3*2) * 2^(2*2))^2 = (5^6 * 2^4)^2

6. (a^4 * b^2)^5:
We apply the exponent to each variable within the parentheses.
(a^4 * b^2)^5 = a^(4*5) * b^(2*5) = a^20 * b^10

7. (3x^5)^3:
We distribute the exponent to both 3 and x^5 within the parentheses.
(3x^5)^3 = 3^3 * (x^5)^3 = 3^3 * x^(5*3) = 27x^15

8. 7^4/7^2:
To simplify, we subtract the exponents since the bases (7) are the same.
7^4/7^2 = 7^(4-2) = 7^2 = 49

9. x^5/x^8:
We subtract the exponents since the bases (x) are the same.
x^5/x^8 = x^(5-8) = x^-3 = 1/x^3

10. 12m^9/4m^5:
We divide the coefficients and subtract the exponents of m.
12m^9/4m^5 = (12/4)(m^(9-5)) = 3m^4

11. (3/4)^2:
We square the numerator and the denominator.
(3/4)^2 = 3^2/4^2 = 9/16

12. (x/y)^5:
We apply the exponent to both x and y within the parentheses.
(x/y)^5 = x^5 / y^5

13. (2x^5/5y^4)^3:
We apply the exponent to both the numerator and the denominator within the parentheses.
(2x^5/5y^4)^3 = (2^3 * (x^5)^3) / (5^3 * (y^4)^3) = 8x^15 / 125y^12

14. (12m^4/6)^3:
We divide the coefficient by the denominator, and apply the exponent to m^4 within the parentheses.
(12m^4/6)^3 = (12/6) * (m^4)^3 = 2m^12

15. (3/9a^3)^4:
We simplify the fraction and apply the exponent to both the numerator and the denominator within the parentheses.
(3/9a^3)^4 = (1/3a^3)^4 = 1^4 / (3^4 * (a^3)^4) = 1 / (81a^12)

B.
1. To simplify each expression, I followed the rules of the laws of exponents. For multiplication, I added the exponents of the same base, and for division, I subtracted the exponents.

2. Understanding the laws of exponents and practicing regularly would make it easier to simplify the given expressions. Familiarity with the rules allows you to quickly identify which operations to apply, saving time and reducing errors. Regular practice helps build confidence and fluency in manipulating exponents.