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We are given the following system of equations for two numbers \( a \) and \( b \):

\[
a + b = 2
\]
\[
a^2 + b^2 = 6
\]

### Part (a): Find the value of \( a^3 + b^3 \).

We can use the identity for the sum of cubes:

\[
a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)
\]

From the first equation, we know:

\[
a + b = 2
\]

Next, to find \( ab \), we use the identity:

\[
a^2 + b^2 = (a + b)^2 - 2ab
\]

Substituting \( a + b = 2 \) and \( a^2 + b^2 = 6 \):

\[
6 = 2^2 - 2ab
\]

\[
6 = 4 - 2ab
\]

\[
2ab = -2
\]

\[
ab = -1
\]

Now, substituting \( a + b = 2 \), \( ab = -1 \), and \( a^2 + b^2 = 6 \) into the expression for \( a^3 + b^3 \):

\[
a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right) = 2 \times (6 - (-1)) = 2 \times 7 = 14
\]

Thus, the value of \( a^3 + b^3 \) is:

\[
a^3 + b^3 = 14
\]

### Part (b): Find the value of \( a^5 + b^5 \).

We will use the identity:

\[
a^5 + b^5 = (a + b)(a^4 - ab(a^3 + b^3) + b^4)
\]

We need to compute \( a^4 + b^4 \). Using the identity for the fourth powers:

\[
a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2
\]

We already know that \( a^2 + b^2 = 6 \) and \( ab = -1 \), so:

\[
a^4 + b^4 = 6^2 - 2(-1)^2 = 36 - 2 = 34
\]

Now, using the expression for \( a^5 + b^5 \):

\[
a^5 + b^5 = (a + b)\left(a^4 - ab(a^3 + b^3) + b^4\right)
\]

Substituting \( a + b = 2 \), \( ab = -1 \), \( a^3 + b^3 = 14 \), and \( a^4 + b^4 = 34 \):

\[
a^5 + b^5 = 2 \times (34 - (-1)(14)) = 2 \times (34 + 14) = 2 \times 48 = 96
\]

Thus, the value of \( a^5 + b^5 \) is:

\[
a^5 + b^5 = 96
\]

### Summary:
- \( a^3 + b^3 = 14 \)
- \( a^5 + b^5 = 96 \)

Would you like more details on any step?

Here are 5 related questions:
1. How would you solve for \( a \) and \( b \) individually in this system?
2. What is the general formula for \( a^n + b^n \) in terms of lower powers of \( a \) and \( b \)?
3. Can you express \( a^7 + b^7 \) using a similar approach?
4. How can these methods be extended to complex numbers?
5. How does the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) generalize?

**Tip**: Using algebraic identities simplifies the computation of higher powers, reducing the need to solve for \( a \) and \( b \) explicitly.

Two numbers a and b satisfy the equations: a + b = 2 and a^2 + b^2 = 6. (a) What is the value of a^3 + b^3? (b) What is the value of a^5 + b^5?

Learn how to determine the values of a^3 + b^3 and a^5 + b^5 given the system of equations a + b = 2 and a^2 + b^2 = 6. This problem involves algebraic identities, polynomial manipulation, and solving systems of equations. Suitable for students in Grades 10-12.

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