To find the value of \(a^3 + b^3\) given that \(a + b = 5\) and \(ab = 13\), we can use the identity for the sum of cubes:
\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\).
First, let's calculate \(a^2 + b^2\) using the formula:
\((a+b)^2 = a^2 + b^2 + 2ab\).
We know:
- \(a + b = 5\)
- \(ab = 13\)
Therefore:
\((a+b)^2 = 5^2 = 25\).
Substitute into the formula:
\(a^2 + b^2 + 2ab = 25\).
Substitute \(ab = 13\):
\(a^2 + b^2 + 2(13) = 25\)
\(a^2 + b^2 + 26 = 25\)
\(a^2 + b^2 = 25 - 26 = -1\).
Now, substitute values into the cube formula:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
\(a^3 + b^3 = 5(a^2 - ab + b^2)\)
Substitute the known values:
\(a^2 + b^2 = -1\), \(ab = 13\)
\(a^2 - ab + b^2 = -1 - 13 = -14\)
\(a^3 + b^3 = 5 \times (-14) = -70\)
Thus, the value of \(a^3 + b^3\) is -70.
The correct option is -70.
Similar Questions
Simplify: $3((\frac{5}{3})x^2 - 28x + 15) - 5(x^2 + 6x - 15)$
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Important Questions from Algebric Equations
If m + n = 24, then (m - 16)³ + (n - 8)³ is ____.
For the following equations, what are the values of a and b to have infinitely many solutions?
ax + by = 2
3x - (5 - 2ay) = 6
If m + n = 24, then (m - 16)³ + (n - 8)³ is ____.
For the following equations, what are the values of a and b to have infinitely many solutions?
ax + by = 2
3x - (5 - 2ay) = 6
Simplify the following expression:
\(\frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{(x - y)(y - z)(z - x)}\)
