\sqrt{\frac{7}{12}} - \sqrt{\frac{1}{3}} = \dots
Answer (1)
To solve the expression 12 7 − 3 1 , let's go through the steps systematically:
Finding a Common Denominator :
The fractions inside the square roots are 12 7 and 3 1 .
To simplify these expressions, we need a common denominator. The least common denominator of 12 and 3 is 12.
Rewrite 3 1 with a denominator of 12: 3 1 = 12 4 .
Express the Square Roots :
Now, we have 12 7 and 12 4 .
Let's simplify these expressions separately under the same denominator.
Simplify Each Square Root :
For 12 7 , it remains as is since 7 is not a perfect square.
For 12 4 , notice that 4 is a perfect square. So, we can simplify it further: 12 4 = 12 4 = 12 4 = 12 2 .
Simplifying 12 2 : 12 2 = 4 × 3 2 = 2 3 2 = 3 1 .
3 1 can be further simplified by rationalizing the denominator: 3 1 = 3 3 .
Combine the Expressions :
Now we have 12 7 − 3 3 .
To simplify further, we may need to express 12 7 in a compatible form for direct subtraction. Since the initial simplification gets quite complex here without direct means, we express both using radicals over a common base directly if necessary or approximate.
Solution :
The exact form is 12 7 − 3 3 .
If approximating values are allowed, numerical calculations can follow depending on context requirements.
This detailed procedure outlines simplifying and manipulating expressions involving square roots, emphasizing the importance of common denominators and rationalizing as needed to reach simplified forms or indicate numeric approximation.
