Question

If \(a=\sqrt{11}+\sqrt{3}, \mathrm{~b}=\sqrt{12}+\sqrt{2}\) and \(\mathrm{c}=\sqrt{6}+\sqrt{4}\), then which of the following holds true? (1) \(c>a>b\) (2) \(a>b>c\) (3) \(\mathrm{a}>\mathrm{c}>\mathrm{b}\) (4) \(b>a>c\)

Short answer

Based on the given expressions and their simplifications, the correct order relation between them is: (2) \(a > b > c\).

Step-by-step solution

Simplifying Expression a

We will start by simplifying the expression given for a: \(a = \sqrt{11} + \sqrt{3}\). As there are no further operations we can do, this is the simplest form for a.

Simplifying Expression b

Now, we will simplify the expression given for b: \(b = \sqrt{12} + \sqrt{2}\). Notice that \(\sqrt{12}\) can be simplified, as 12 can be expressed as \(2 \times 2 \times 3\). So we have: \(b = \sqrt{2 \times 2 \times 3} + \sqrt{2}\). Now, we can simplify the square root of 12 using the properties of square roots: \(b = 2\sqrt{3} + \sqrt{2}\).

Simplifying Expression c

Finally, we will simplify the expression given for c: \(c = \sqrt{6} + \sqrt{4}\). Notice that \(\sqrt{4}\) is a perfect square and can be simplified to 2. So now we have: \(c = \sqrt{6} + 2\).

Comparing the Simplified Expressions

Now, we will compare the simplified expressions for a, b, and c. From the simplified expressions, we can see that - a involves \(\sqrt{11}+\sqrt{3}\), - b includes \(2\sqrt{3}+\sqrt{2}\), and - c has \(\sqrt{6} + 2\). Since \(\sqrt{11} > \sqrt{6}\) and \(\sqrt{3} > \sqrt{2}\), we can conclude that \(a > b\). Between a and c, we have \(\sqrt{11} > \sqrt{6}\), and on the other hand, \(2 > \sqrt{3}\). To determine the relationship between a and c, we can compare the sum of the square roots of 11 and 6 with 2. As \(\sqrt{11} + \sqrt{6} > 2\), we can conclude that \(a > c\). Now we know that \(a > b\) and \(a > c\). Comparing b and c, we have \(2\sqrt{3} > \sqrt{6}\) and \(\sqrt{2} > 2\). Since \(2\sqrt{3} + \sqrt{2} > \sqrt{6} + 2\), we can conclude that \(b > c\). Finally, we obtain \(a > b > c\). Therefore, the correct answer is: (2) \(a>b>c\).

Key concepts

Inequalities

Inequalities are mathematical expressions indicating that one quantity is larger or smaller than another. They involve symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). To determine inequalities, we can compare numerical values directly or evaluate expressions to deduce their relative sizes.When comparing expressions containing surds—square roots or other irrational numbers—we rely on understanding their approximate values without necessarily calculating the exact numbers. It's crucial to ensure all expressions are simplified equivalently, as shown in the exercise, before making a fair comparison. In this problem, we utilized simplification to make clearer comparisons and deduced that given expressions satisfy the inequality:

• \(a > b\)
• \(b > c\)

Thus, we concluded \(a > b > c\), illustrating how inequalities can order various mathematical expressions.

Simplification of expressions

Simplification is the process of transforming complex expressions into simpler or more manageable forms. Here, it involves "collecting like terms," breaking down numbers inside square roots to more easily comparable forms, and using algebraic identities.In the example given:

• \(a = \sqrt{11} + \sqrt{3}\), didn't need further simplification since it was already in its simplest form.
• \(b = \sqrt{12} + \sqrt{2}\), where \(\sqrt{12}\) was simplified to \(2\sqrt{3}\) using factors, resulting in \(b = 2\sqrt{3} + \sqrt{2}\).
• \(c = \sqrt{6} + \sqrt{4}\), was simplified by turning the perfect square \(\sqrt{4}\) into 2, leading to \(c = \sqrt{6} + 2\).

These steps helped us achieve a state where direct comparisons using inequalities became straightforward. Simplification, therefore, is a key skill in algebra for solving and understanding expressions effectively.

Comparison of surds

Surds refer to irrational numbers that remain in the form of roots, such as square roots, cube roots, etc. When comparing surds, exact values are often not calculated; rather, approximations or properties of numbers are considered.Important strategies for comparing surds include square root simplifications and recognizing numerical orders (for example, knowing that \(\sqrt{11} > \sqrt{6}\) since 11 is greater than 6). In this exercise:

• We compared \(\sqrt{11}\), \(\sqrt{3}\), \(\sqrt{12}\), \(\sqrt{2}\), and \(\sqrt{4}\) by their simplified forms.
• Each expression was broken down into realizable terms or combinations, like turning \(\sqrt{12}\) into \(2\sqrt{3}\), to make comparison easier.
• By evaluating each simplified expression, the comparison led us to find that \(a > b > c\).

Comparing surds efficiently requires careful observation and sometimes involves innovative simplification or approximations to establish meaningful numeric relationships.