Find the value of (−5√3 + 3√12 + 2√75) in the form of (a + b√3), where a and b are rational numbers.

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Points to Remember:

  • Simplify each term within the parentheses.
  • Express each term as a multiple of √3.
  • Combine like terms to obtain a simplified expression in the form a + b√3.
  • Identify the rational numbers ‘a’ and ‘b’.

Introduction:

This question requires simplifying a mathematical expression involving square roots. The expression (−5√3 + 3√12 + 2√75) needs to be reduced to its simplest form, which is expressed as (a + b√3), 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:

  • -5√3: This term is already in its simplest form.

  • 3√12: We can simplify √12 as follows: √12 = √(4 x 3) = √4 x √3 = 2√3. Therefore, 3√12 = 3(2√3) = 6√3.

  • 2√75: We can simplify √75 as follows: √75 = √(25 x 3) = √25 x √3 = 5√3. Therefore, 2√75 = 2(5√3) = 10√3.

2. Combining Like Terms:

Now, let’s substitute the simplified terms back into the original expression:

(−5√3 + 3√12 + 2√75) = (−5√3 + 6√3 + 10√3)

Combining the terms with √3, we get:

(−5 + 6 + 10)√3 = 11√3

3. Expressing in the Required Form:

The simplified expression is 11√3. This is in the form (a + b√3), where a = 0 and b = 11. Both ‘a’ and ‘b’ are rational numbers.

Conclusion:

The value of (−5√3 + 3√12 + 2√75) simplifies to 11√3. This is expressed in the form (a + b√3), 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.

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