Question

. Which of the following are rational and which are irrational? (a) \(-\sqrt{9}\) (b) \(0.375\) (c) \((3 \sqrt{2})(5 \sqrt{2})\) (d) \((1+\sqrt{3})^{2}\)

Short answer

(a) Rational, (b) Rational, (c) Rational, (d) Irrational.

Step-by-step solution

Evaluate and Categorize (a)

Given expression: \(-\sqrt{9}\)First, calculate \(\sqrt{9}\), which is 3. Then, apply the negative sign: \(-\sqrt{9} = -3\).Since \(-3\) can be expressed as the fraction \(-\frac{3}{1}\), it is a rational number.

Evaluate and Categorize (b)

Given number:\(0.375\)A number is rational if it can be written as a fraction. Convert 0.375 to a fraction:\(0.375 = \frac{375}{1000}\).Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, 125:\(\frac{375}{1000} = \frac{3}{8}\).Since \(\frac{3}{8}\) is a fraction, \(0.375\) is a rational number.

Evaluate and Categorize (c)

Given expression:\((3 \sqrt{2})(5 \sqrt{2})\)Simplify the expression by multiplying the terms:\(3 \cdot 5 \cdot \sqrt{2} \cdot \sqrt{2} = 15 \cdot 2 = 30\).Since 30 is an integer and can be expressed as a fraction (\(\frac{30}{1}\)), it is a rational number.

Evaluate and Categorize (d)

Given expression:\((1+\sqrt{3})^{2}\)Expand the expression using the formula \((a+b)^{2} = a^{2} + 2ab + b^{2}\):\((1+\sqrt{3})^{2} = 1^{2} + 2 \times 1 \times \sqrt{3} + (\sqrt{3})^{2}\).Calculate each part individually:- \(1^{2} = 1\)- \(2 \times 1 \times \sqrt{3} = 2\sqrt{3}\)- \((\sqrt{3})^{2} = 3\)Combine the results:\(1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}\).The term \(2\sqrt{3}\) is irrational, thus the expression results in an irrational number.

Key concepts

Square Roots

Square roots are numbers that, when multiplied by themselves, give the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 is 9. Not all numbers have whole numbers as their square roots; some have decimal or fractional values. These can sometimes be irrational, meaning they can't be precisely written as fractions. When you encounter square roots in calculations, it's important to check whether they simplify to a whole number or remain in their root form. For instance:

• The square root of 4 is 2, which is a rational number.
• The square root of 2 cannot be simplified into a whole number and is considered irrational.

Remember to handle negative signs carefully while dealing with square roots, as they impact the final result.

Fraction Conversion

Converting a number into a fraction is a crucial skill, as many numbers can be expressed this way. When working with decimals like 0.375, you can convert them into fractions to examine their properties. To convert:

• Consider the decimal part as the numerator.
• The denominator is usually a power of ten, depending on how many decimal places there are.

For example, 0.375 can be written as \(\frac{375}{1000}\). It's essential to simplify it by finding the greatest common divisor of both the numerator and the denominator. For 0.375, dividing both by 125 gives \(\frac{3}{8}\), a much simpler fraction. Simplified fractions make it easier to determine if a number is rational.

Simplification Methods

Simplification is all about making expressions or numbers as simple as possible while keeping their meaning intact. Often, expressions include square roots or complex terms that need simplifying. Like the expression \((3 \sqrt{2})(5 \sqrt{2})\), simplification involves combining like terms:

• Multiply numbers outside the root together.
• Multiply numbers inside the root together separately.

This specific example becomes: \[3 \times 5 \times \sqrt{2 \times 2} = 15 \times 2 = 30\]Simplifying helps determine whether the resulting number is rational or not.

Expression Expansion

Expression expansion involves using formulas or rules to expand expressions, especially those using parentheses. The expression \((1 + \sqrt{3})^{2}\) can be expanded by using the binomial expansion formula \((a + b)^{2} = a^{2} + 2ab + b^{2}\):

• First term: Square the first number, \(1^{2}\), resulting in 1.
• Second term: Multiply both numbers by 2, \(2 \times 1 \times \sqrt{3}\), resulting in \(2\sqrt{3}\).
• Third term: Square the second number, \((\sqrt{3})^{2}\), resulting in 3.

Combine these results to get the expanded expression: \( 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}\).While the constant parts add up to 4, it is the \(2\sqrt{3}\) that introduces irrationality since \(\sqrt{3}\) itself is an irrational number. So the whole expression results in an irrational number, owing to its irrational component.