Let's evaluate each expression one by one in detail.
### Expression 5: [tex]\( g^5 - h^3 \)[/tex] where [tex]\( g = 2 \)[/tex] and [tex]\( h = 7 \)[/tex]
1. Calculate [tex]\( g^5 \)[/tex]:
[tex]\[
g^5 = 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32
\][/tex]
2. Calculate [tex]\( h^3 \)[/tex]:
[tex]\[
h^3 = 7^3 = 7 \times 7 \times 7 = 343
\][/tex]
3. Subtract [tex]\( h^3 \)[/tex] from [tex]\( g^5 \)[/tex]:
[tex]\[
g^5 - h^3 = 32 - 343 = -311
\][/tex]
So, [tex]\( g^5 - h^3 = -311 \)[/tex].
### Expression 10: [tex]\( c^2 + d^3 \)[/tex] where [tex]\( c = 8 \)[/tex] and [tex]\( d = -3 \)[/tex]
1. Calculate [tex]\( c^2 \)[/tex]:
[tex]\[
c^2 = 8^2 = 8 \times 8 = 64
\][/tex]
2. Calculate [tex]\( d^3 \)[/tex]:
[tex]\[
d^3 = (-3)^3 = -3 \times -3 \times -3 = -27
\][/tex]
3. Add [tex]\( c^2 \)[/tex] and [tex]\( d^3 \)[/tex]:
[tex]\[
c^2 + d^3 = 64 + (-27) = 64 - 27 = 37
\][/tex]
So, [tex]\( c^2 + d^3 = 37 \)[/tex].
### Expression 11: [tex]\( a^2 \cdot b^6 \)[/tex] where [tex]\( a = \frac{1}{2} \)[/tex] and [tex]\( b = 2 \)[/tex]
1. Calculate [tex]\( a^2 \)[/tex]:
[tex]\[
a^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\][/tex]
2. Calculate [tex]\( b^6 \)[/tex]:
[tex]\[
b^6 = 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64
\][/tex]
3. Multiply [tex]\( a^2 \)[/tex] and [tex]\( b^6 \)[/tex]:
[tex]\[
a^2 \cdot b^6 = \frac{1}{4} \times 64 = 16
\][/tex]
So, [tex]\( a^2 \cdot b^6 = 16 \)[/tex].
### Expression 12: [tex]\( (r - s)^3 + r^2 \)[/tex] where [tex]\( r = -3 \)[/tex] and [tex]\( s = -4 \)[/tex]
1. Calculate the difference [tex]\( r - s \)[/tex]:
[tex]\[
r - s = -3 - (-4) = -3 + 4 = 1
\][/tex]
2. Calculate [tex]\( (r - s)^3 \)[/tex]:
[tex]\[
(r - s)^3 = 1^3 = 1
\][/tex]
3. Calculate [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = (-3)^2 = (-3) \times (-3) = 9
\][/tex]
4. Add [tex]\( (r - s)^3 \)[/tex] and [tex]\( r^2 \)[/tex]:
[tex]\[
(r - s)^3 + r^2 = 1 + 9 = 10
\][/tex]
So, [tex]\( (r - s)^3 + r^2 = 10 \)[/tex].
In summary, the evaluated results for each expression are:
1. [tex]\( g^5 - h^3 = -311 \)[/tex]
2. [tex]\( c^2 + d^3 = 37 \)[/tex]
3. [tex]\( a^2 \cdot b^6 = 16 \)[/tex]
4. [tex]\( (r - s)^3 + r^2 = 10 \)[/tex]
