Certainly! Let's solve each part step-by-step:
### Part (a)
Expression: [tex]\((3)^2 \times (3)^6\)[/tex]
1. Combine the exponents:
[tex]\[
3^2 \times 3^6 = 3^{2+6} = 3^8
\][/tex]
2. Calculate the power:
[tex]\[
3^8 = 6561
\][/tex]
Thus, [tex]\((3)^2 \times (3)^6 = 6561\)[/tex].
### Part (b)
Expression: [tex]\(\left(\frac{5}{2}\right)^6 \times \left(\frac{5}{2}\right)^2\)[/tex]
1. Combine the exponents:
[tex]\[
\left(\frac{5}{2}\right)^6 \times \left(\frac{5}{2}\right)^2 = \left(\frac{5}{2}\right)^{6+2} = \left(\frac{5}{2}\right)^8
\][/tex]
2. Calculate the power:
[tex]\[
\left(\frac{5}{2}\right)^8 = 1525.87890625
\][/tex]
Thus, [tex]\(\left(\frac{5}{2}\right)^6 \times \left(\frac{5}{2}\right)^2 = 1525.87890625\)[/tex].
### Part (c)
Expression: [tex]\(\left(\frac{7}{4}\right)^{10} \times \left(\frac{7}{4}\right)^{-3}\)[/tex]
1. Combine the exponents:
[tex]\[
\left(\frac{7}{4}\right)^{10} \times \left(\frac{7}{4}\right)^{-3} = \left(\frac{7}{4}\right)^{10-3} = \left(\frac{7}{4}\right)^7
\][/tex]
2. Calculate the power:
[tex]\[
\left(\frac{7}{4}\right)^7 = 50.26507568359375
\][/tex]
Thus, [tex]\(\left(\frac{7}{4}\right)^{10} \times \left(\frac{7}{4}\right)^{-3} = 50.26507568359375\)[/tex].
### Part (d)
Expression: [tex]\(\left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^3\)[/tex]
1. Combine the exponents:
[tex]\[
\left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^3 = \left(\frac{1}{2}\right)^{5+3} = \left(\frac{1}{2}\right)^8
\][/tex]
2. Calculate the power:
[tex]\[
\left(\frac{1}{2}\right)^8 = 0.00390625
\][/tex]
Thus, [tex]\(\left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^3 = 0.00390625\)[/tex].
In summary:
- [tex]\((3)^2 \times (3)^6 = 6561\)[/tex]
- [tex]\(\left(\frac{5}{2}\right)^6 \times \left(\frac{5}{2}\right)^2 = 1525.87890625\)[/tex]
- [tex]\(\left(\frac{7}{4}\right)^{10} \times \left(\frac{7}{4}\right)^{-3} = 50.26507568359375\)[/tex]
- [tex]\(\left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^3 = 0.00390625\)[/tex]
