If \(x + \left( {\frac{1}{x}} \right) = 12\) and \({x^2} - \frac{1}{{{x^2}}} = 50\) , then the value of \({x^4} - \frac{1}{{{x^4}}} \) is:
7100
Finding the Value of \(x^4 - \frac{1}{x^4}\) Using Algebraic Identities
The problem asks us to find the value of the expression \(x^4 - \frac{1}{x^4}\) given two initial conditions: \(x + \left( {\frac{1}{x}} \right) = 12\) and \({x^2} - \frac{1}{{{x^2}}} = 50\).
Let's analyze the expression we need to find, \(x^4 - \frac{1}{x^4}\). This expression can be factored as a difference of squares:
\[x^4 - \frac{1}{x^4} = \left(x^2\right)^2 - \left(\frac{1}{x^2}\right)^2\]Using the algebraic identity \(a^2 - b^2 = (a-b)(a+b)\), where \(a = x^2\) and \(b = \frac{1}{x^2}\), we get:
\[x^4 - \frac{1}{x^4} = \left(x^2 - \frac{1}{x^2}\right) \left(x^2 + \frac{1}{x^2}\right)\]We are already given the value of one of the factors:
- \({x^2} - \frac{1}{{{x^2}}} = 50\)
Now, we need to find the value of the other factor, \(x^2 + \frac{1}{x^2}\). We can find this using the first given condition: \(x + \left( {\frac{1}{x}} \right) = 12\).
Let's square both sides of the equation \(x + \frac{1}{x} = 12\):
\[\left(x + \frac{1}{x}\right)^2 = 12^2\]Using the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = x\) and \(b = \frac{1}{x}\), we get:
\[x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = 144\]Simplify the middle term:
\[x^2 + 2 + \frac{1}{x^2} = 144\]Now, isolate \(x^2 + \frac{1}{x^2}\) by subtracting 2 from both sides:
\[x^2 + \frac{1}{x^2} = 144 - 2\] \[x^2 + \frac{1}{x^2} = 142\]So, we have found the value of the second factor:
- \(x^2 + \frac{1}{x^2} = 142\)
Now we have both parts needed to calculate \(x^4 - \frac{1}{x^4}\):
\[x^4 - \frac{1}{x^4} = \left(x^2 - \frac{1}{x^2}\right) \left(x^2 + \frac{1}{x^2}\right)\]Substitute the values we found:
\[x^4 - \frac{1}{x^4} = (50) \times (142)\]Perform the multiplication:
\[50 \times 142 = 50 \times (100 + 40 + 2) = 5000 + 2000 + 100 = 7100\]Thus, the value of \(x^4 - \frac{1}{x^4}\) is 7100.
Step-by-Step Calculation Summary
| Step | Description | Calculation |
|---|---|---|
| 1 | Identify the target expression structure | \(x^4 - \frac{1}{x^4} = \left(x^2 - \frac{1}{x^2}\right) \left(x^2 + \frac{1}{x^2}\right)\) |
| 2 | Use the given \(x^2 - \frac{1}{x^2}\) value | \({x^2} - \frac{1}{{{x^2}}} = 50\) |
| 3 | Use the given \(x + \frac{1}{x}\) to find \(x^2 + \frac{1}{x^2}\) | \(\left(x + \frac{1}{x}\right)^2 = 12^2 \implies x^2 + 2 + \frac{1}{x^2} = 144 \implies x^2 + \frac{1}{x^2} = 142\) |
| 4 | Multiply the two factors | \(x^4 - \frac{1}{x^4} = (50) \times (142)\) |
| 5 | Final calculation | \(50 \times 142 = 7100\) |
Revision Table: Key Concepts
| Concept | Description | Relevant Identity |
|---|---|---|
| Difference of Squares | Factoring an expression of the form \(a^2 - b^2\) | \(a^2 - b^2 = (a-b)(a+b)\) |
| Squaring a Binomial Sum | Expanding an expression of the form \((a+b)^2\) | \((a+b)^2 = a^2 + 2ab + b^2\) |
| Algebraic Manipulation | Rearranging equations to isolate desired terms | e.g., \(x^2 + 2 + \frac{1}{x^2} = 144 \implies x^2 + \frac{1}{x^2} = 142\) |
Additional Information on Algebraic Identities
Algebraic identities are equations that are true for all values of the variables involved. They are powerful tools for simplifying expressions and solving equations. In this problem, we used two common identities:
- Difference of Squares: \(a^2 - b^2 = (a-b)(a+b)\). This identity is crucial for breaking down higher powers into products of lower powers, as seen with \(x^4 - \frac{1}{x^4}\).
- Perfect Square Trinomials: \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\). These are useful for relating expressions like \(x + \frac{1}{x}\) to \(x^2 + \frac{1}{x^2}\) or \(x^2 - \frac{1}{x^2}\). In this case, squaring \(x + \frac{1}{x}\) conveniently gave us a term \(x \cdot \frac{1}{x}\) which simplifies to 1, helping us find \(x^2 + \frac{1}{x^2}\).
Understanding and recognizing these patterns is key to solving many algebraic problems efficiently.
Important Questions from Identities
(x - y) 3+ (y - z) 3+ (z - x) 3= ?
- \((\sqrt{7} + \sqrt{9})(\sqrt{7} - \sqrt{9})\) is equal to:
If x satisfies the equation x 2 - 2x + 1 = 0, then the value of \(\rm x^3 - \frac{1}{x^3}\) is:
If x + y = 5 and xy = 6, then find x 3+ y 3
If \(x = \sqrt3 + \sqrt2,\) then the value of \(x^2 + \frac{1}{x^2}\) is:
