We are asked to find the value of the expression $\left(\frac{x^2 + y^2 - xy}{x^3 + y^3}\right)$ given $x = 12$ and $y = 7$.
Evaluating the Expression with Given Values
Substitute the values $x=12$ and $y=7$ into the expression.
Numerator: $x^2 + y^2 - xy$
$12^2 + 7^2 - (12)(7)$
$144 + 49 - 84$
$193 - 84 = 109$
Using Algebraic Factorization for Denominator
The denominator is $x^3 + y^3$. We can use the sum of cubes factorization formula:
$x^3 + y^3 = (x+y)(x^2 - xy + y^2)$
Notice that the term $(x^2 - xy + y^2)$ is exactly the numerator we calculated.
Simplifying the Expression
Now substitute the calculated numerator and the factored denominator back into the original expression:
$\left(\frac{x^2 + y^2 - xy}{x^3 + y^3}\right) = \left(\frac{109}{(x+y)(x^2 - xy + y^2)}\right)$
Substitute the value of the numerator $(109)$ and the value of $(x+y)$:
$x+y = 12 + 7 = 19$
So the expression becomes:
$\frac{109}{(19)(109)}$
Cancel out the common factor $109$ from the numerator and the denominator:
$\frac{1}{19}$
Final Value Calculation
The value of the expression is $\frac{1}{19}$.
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Then the value of $\frac{3x}{2x^2 + 2 - 5x}$ will be: - If $3a + 4b = 2$ and $ab = \frac{1}{36}$, then $27a^3 + 64b^3$ is:
Important Questions from Algebra
If 2x – y = 2 and xy = \(\frac{3}{2}\) , then what is the value of x 3– \(\frac{{{y^3}}}{8}\) ?
If (10a 3+ 4b 3) : (11a 3- 15b 3) = 7 : 5, then (3a + 5b) : (9a - 2b) =?
- If 4sin 2 θ = 3(1+ cos θ), 0° < θ < 90°, then what is the value of (2tan θ + 4sin θ - sec θ)?
The value of:
\(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)
If (x + y) 3+ 27(x - y) 3= (Ax - 2y)(Bx 2+ Cxy + 13y 2), then the value of A - B - C is:
