Finding the Value of $\mathbf{a^3 + b^3}$ Using Algebraic Identities

This problem asks us to find the value of the expression $\(a^3 + b^3\)$ given the sum of two variables, \(a+b\), and their product, \(ab\). We are given that \(a+b = 5\) and \(ab = 6\). To solve this, we can use a known algebraic identity that relates the sum of cubes to the sum and product of the variables.

Understanding the Problem: Given Information

• Sum of \(a\) and \(b\): $\(a+b = 5\)$
• Product of \(a\) and \(b\): $\(ab = 6\)$

We need to calculate the value of $\(a^3 + b^3\)$.

Using the Sum of Cubes Identity for $\mathbf{a^3 + b^3}$

The primary algebraic identity for the sum of cubes is:

$\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)$

We already know the values for $\(a+b\)$ and $\(ab\)$. However, the identity requires the term $\(a^2 + b^2)\)$. We can find the value of $\(a^2 + b^2)\)$ using another common identity involving $\(a+b\)$ and $\(ab\)$.

Finding $\mathbf{a^2 + b^2}$ from $\mathbf{(a+b)^2}$

The identity for the square of a sum is:

$\((a+b)^2 = a^2 + 2ab + b^2\)$

We can rearrange this identity to solve for $\(a^2 + b^2)\)$:

$\(a^2 + b^2 = (a+b)^2 - 2ab\)$

Now, substitute the given values $\(a+b = 5\)$ and $\(ab = 6\)$ into this rearranged identity:

$\(a^2 + b^2 = (5)^2 - 2(6)\)$

$\(a^2 + b^2 = 25 - 12\)$

$\(a^2 + b^2 = 13\)$

So, the value of $\(a^2 + b^2\)$ is 13.

Calculating the Value of $\mathbf{a^3 + b^3}$

Now that we have the values for $\(a+b\)$, $\(ab\)$, and $\(a^2 + b^2)\)$, we can substitute them back into the sum of cubes identity $\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)$.

Substitute the values:

• $\(a+b = 5\)$
• $\(a^2 + b^2 = 13\)$
• $\(ab = 6\)$

Plugging these into the identity:

$\(a^3 + b^3 = (a+b)((a^2 + b^2) - ab)\)$

$\(a^3 + b^3 = (5)(13 - 6)\)$

$\(a^3 + b^3 = (5)(7)\)$

$\(a^3 + b^3 = 35\)$

Therefore, the value of $\(a^3 + b^3\)$ is 35.

Summary of Steps to Find $\mathbf{a^3 + b^3}$

1. Identify the given values: $\(a+b = 5\)$ and $\(ab = 6\)$.
2. Recall the identity for the sum of cubes: $\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)$.
3. Realize the need for $\(a^2 + b^2)\)$. Use the identity $\((a+b)^2 = a^2 + 2ab + b^2\)$ to find $\(a^2 + b^2 = (a+b)^2 - 2ab\)$.
4. Substitute the given values into the $\(a^2 + b^2)\)$ formula: $\(a^2 + b^2 = (5)^2 - 2(6) = 25 - 12 = 13\)$.
5. Substitute the values of $\(a+b\)$, $\(ab\)$, and $\(a^2 + b^2\)$ into the sum of cubes identity: $\(a^3 + b^3 = (5)(13 - 6) = 5 \times 7 = 35\)$.
6. The final value of $\(a^3 + b^3\)$ is 35.

Revision Table: Key Algebraic Identities

Additional Information: Alternative Method for $\mathbf{a^3 + b^3}$

There is another useful identity for the sum of cubes that can directly use the values of $\(a+b\)$ and $\(ab\)$:

$\(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\)$

Let's verify the result using this alternative identity:

$\(a^3 + b^3 = (5)^3 - 3(6)(5)\)$

$\(a^3 + b^3 = 125 - 3 \times 30\)$

$\(a^3 + b^3 = 125 - 90\)$

$\(a^3 + b^3 = 35\)$

Both identities yield the same result, confirming our calculation. This alternative method is often faster when $\(a+b\)$ and $\(ab\)$ are directly given.