Please simplify and explain to me how to do each of these. Greatly appreciated.

x = multiply sign (not a variable)

a)((9)^(1/2) (4)^2) / 24

b) (16)(2)^(x+2) The x here is a variable, not multiplication sign)

c) (3)^(-8)/ (3)^(-6) (3)^(-5)

d) ((4)^(-2) + (2)^(-4))^(-1)

Thank you so much if you can.

((9)^(1/2) (4)^2) / 24

(3 * 16)/24
48/24
2

(16)(2)^(x+2)
(2^4)(2^(x+2)
2^(x+6)
or
64*2^x

(3)^(-8)/ (3)^(-6) (3)^(-5)
(3)^(-8)/ (3)^(-11)
3^(-8-(-11))
3^3
27

((4)^(-2) + (2)^(-4))^(-1)
(1/16 + 1/16)^(-1)
(1/8)^(-1)
8

Thank yu so much!

a) To simplify and solve this expression, we follow the order of operations which is commonly known as PEMDAS (Parentheses, Exponents, Multiplication & Division, and Addition & Subtraction).

1. Start by simplifying the exponents:
(9)^(1/2) = √9 = 3
(4)^2 = 4 × 4 = 16

2. Multiply the simplified values:
3 × 16 = 48

3. Divide the result by 24:
48 ÷ 24 = 2

Therefore, the value of expression a) is 2.

b) For this expression, we need to use the exponent properties. When multiplying exponential expressions with the same base, we add the exponents.

1. Start by simplifying the exponent:
2^(x + 2)

2. Expand the expression and simplify further:
2^x × 2^2
2^x × 4

So, the simplified expression for b) is 4 × 2^x.

c) To simplify this expression, we can use the exponent property that states when dividing exponential expressions with the same base, we subtract the exponents.

1. Start by simplifying the exponents:
(3)^(-8) = 1/(3)^8
(3)^(-6) = 1/(3)^6
(3)^(-5) = 1/(3)^5

2. Substitute the values back into the expression:
(1/(3)^8) ÷ (1/(3)^6) × (1/(3)^5)

3. Simplify further by subtracting the exponents:
(3)^(8-6+5)
(3)^7

Therefore, the simplified value of expression c) is (3)^7.

d) To simplify this expression, we follow the order of operations once again:

1. Start by simplifying the exponents:
(4)^(-2) = 1/(4)^2 = 1/16
(2)^(-4) = 1/(2)^4 = 1/16

2. Add the fractions:
1/16 + 1/16 = 2/16

3. Invert the sum to find the reciprocal:
(2/16)^(-1) = 16/2

4. Simplify further:
16/2 = 8

Thus, the value of expression d) is 8.