-2
Understanding the Problem: Evaluating a Radical Expression
The question asks us to find the value of the expression \((\sqrt{7} + \sqrt{9})(\sqrt{7} - \sqrt{9})\). This expression involves square roots and looks like a specific algebraic form that can be simplified easily.
Applying the Difference of Squares Identity
The given expression \((\sqrt{7} + \sqrt{9})(\sqrt{7} - \sqrt{9})\) is in the form \((a+b)(a-b)\). There is a fundamental algebraic identity for this form, known as the difference of squares identity:
\[(a+b)(a-b) = a^2 - b^2\]
We can apply this identity directly to simplify the given expression. In this expression:
- \(a = \sqrt{7}\)
- \(b = \sqrt{9}\)
Step-by-Step Calculation
Using the difference of squares identity \((a+b)(a-b) = a^2 - b^2\), we substitute \(a = \sqrt{7}\) and \(b = \sqrt{9}\) into the formula:
\[(\sqrt{7} + \sqrt{9})(\sqrt{7} - \sqrt{9}) = (\sqrt{7})^2 - (\sqrt{9})^2\]
Now, we need to evaluate the squares of the square roots:
- The square of a square root cancels out the square root symbol. So, \((\sqrt{7})^2 = 7\).
- Similarly, \((\sqrt{9})^2 = 9\). Note that \(\sqrt{9}\) is also equal to 3, so \(3^2 = 9\). Both ways yield 9.
Substitute these values back into the expression:
\[(\sqrt{7})^2 - (\sqrt{9})^2 = 7 - 9\]
Finally, perform the subtraction:
\[7 - 9 = -2\]
Thus, the value of the expression \((\sqrt{7} + \sqrt{9})(\sqrt{7} - \sqrt{9})\) is -2.
Summarizing the Solution
Here is a quick summary of the steps:
- Recognize the expression \((\sqrt{7} + \sqrt{9})(\sqrt{7} - \sqrt{9})\) as the form \((a+b)(a-b)\).
- Apply the difference of squares identity: \((a+b)(a-b) = a^2 - b^2\).
- Substitute \(a=\sqrt{7}\) and \(b=\sqrt{9}\): \((\sqrt{7})^2 - (\sqrt{9})^2\).
- Evaluate the squares: \(7 - 9\).
- Calculate the final value: \(-2\).
| Step | Description | Calculation |
|---|---|---|
| 1 | Identify form | \((a+b)(a-b)\) |
| 2 | Apply identity | \(a^2 - b^2\) |
| 3 | Substitute values | \((\sqrt{7})^2 - (\sqrt{9})^2\) |
| 4 | Evaluate squares | \(7 - 9\) |
| 5 | Final calculation | \(-2\) |
Revision Table: Key Concepts in Algebraic Identities and Square Roots
| Concept | Description | Example |
|---|---|---|
| Difference of Squares Identity | \((a+b)(a-b) = a^2 - b^2\) | \((x+2)(x-2) = x^2 - 4\) |
| Square of a Square Root | \((\sqrt{x})^2 = x\) for \(x \ge 0\) | \((\sqrt{5})^2 = 5\), \((\sqrt{16})^2 = 16\) |
| Square Root of a Perfect Square | \(\sqrt{x^2} = |x|\). If \(x \ge 0\), \(\sqrt{x^2}=x\). | \(\sqrt{9} = \sqrt{3^2} = 3\), \(\sqrt{25} = \sqrt{5^2} = 5\) |
Additional Information: Why the Difference of Squares Identity Works
The difference of squares identity \((a+b)(a-b) = a^2 - b^2\) can be derived by simply expanding the left side using the distributive property (also known as FOIL):
- First terms: \(a \times a = a^2\)
- Outer terms: \(a \times (-b) = -ab\)
- Inner terms: \(b \times a = +ab\)
- Last terms: \(b \times (-b) = -b^2\)
Adding these terms together:
\[a^2 - ab + ab - b^2\]
The middle terms \(-ab\) and \(+ab\) cancel each other out (\(-ab + ab = 0\)), leaving:
\[a^2 - b^2\]
This shows why the identity holds true and is very useful for simplifying expressions like the one in the question, especially when dealing with square roots, as squaring a square root simplifies it to the number inside.
Important Questions from Identities
(x - y) 3+ (y - z) 3+ (z - x) 3= ?
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:
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:
