If a + b = 5 and ab = 6, then what is the value of a 3+ b 3?
35
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 $\mathbf{a^3 + b^3}$
- Identify the given values: $\(a+b = 5\)$ and $\(ab = 6\)$.
- Recall the identity for the sum of cubes: $\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)$.
- Realize the need for $\(a^2 + b^2)\)$. Use the identity $\((a+b)^2 = a^2 + 2ab + b^2\)$ to find $\(a^2 + b^2 = (a+b)^2 - 2ab\)$.
- Substitute the given values into the $\(a^2 + b^2)\)$ formula: $\(a^2 + b^2 = (5)^2 - 2(6) = 25 - 12 = 13\)$.
- Substitute the values of $\(a+b\)$, $\(ab\)$, and $\(a^2 + b^2\)$ into the sum of cubes identity: $\(a^3 + b^3 = (5)(13 - 6) = 5 \times 7 = 35\)$.
- The final value of $\(a^3 + b^3\)$ is 35.
Revision Table: Key Algebraic Identities
| Identity Name | Formula |
|---|---|
| Sum of Squares (derived) | $\(a^2 + b^2 = (a+b)^2 - 2ab\)$ |
| Sum of Cubes | $\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)$ |
| Sum of Cubes (alternative form) | $\(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\)$ |
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.
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