Points to Remember:
- Simplify each term within the parentheses.
- Express each term as a multiple of â3.
- Combine like terms to obtain a simplified expression in the form a + bâ3.
- Identify the rational numbers ‘a’ and ‘b’.
Introduction:
This question requires simplifying a mathematical expression involving square roots. The expression (â5â3 + 3â12 + 2â75) needs to be reduced to its simplest form, which is expressed as (a + bâ3), where ‘a’ and ‘b’ are rational numbers (numbers that can be expressed as a fraction of two integers). This involves understanding the properties of square roots and simplifying radicals.
Body:
1. Simplifying Individual Terms:
Let’s simplify each term separately:
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-5â3: This term is already in its simplest form.
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3â12: We can simplify â12 as follows: â12 = â(4 x 3) = â4 x â3 = 2â3. Therefore, 3â12 = 3(2â3) = 6â3.
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2â75: We can simplify â75 as follows: â75 = â(25 x 3) = â25 x â3 = 5â3. Therefore, 2â75 = 2(5â3) = 10â3.
2. Combining Like Terms:
Now, let’s substitute the simplified terms back into the original expression:
(â5â3 + 3â12 + 2â75) = (â5â3 + 6â3 + 10â3)
Combining the terms with â3, we get:
(â5 + 6 + 10)â3 = 11â3
3. Expressing in the Required Form:
The simplified expression is 11â3. This is in the form (a + bâ3), where a = 0 and b = 11. Both ‘a’ and ‘b’ are rational numbers.
Conclusion:
The value of (â5â3 + 3â12 + 2â75) simplifies to 11â3. This is expressed in the form (a + bâ3), where a = 0 and b = 11, both of which are rational numbers. The simplification process involved breaking down the square roots into their simplest forms and then combining like terms. This demonstrates a fundamental understanding of simplifying radical expressions, a crucial skill in algebra and related mathematical fields. No further policy recommendations or best practices are needed as this is a purely mathematical problem. The solution showcases the elegance and precision inherent in mathematical operations.