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Chapter 8: Problem 127

How would you make the numerator of \(\frac{\sqrt{3}}{2}\) a rational number?

Short Answer

Expert verified
Multiply the given fraction by \(\frac{\sqrt{3}}{\sqrt{3}}\), then multiply the resultant fraction by \(\frac{\sqrt{3}}{\sqrt{3}}\) again. As a result, \(\frac{\sqrt{3}}{2}\) becomes \(\frac{\sqrt{3}}{2}\) where the numerator is a rational number. This form retains the value of the original fraction but now has a rational numerator and a rational denominator.

Step by step solution

01

Comprehend the Problem

Recognize that \(\sqrt{3}\) is an irrational number and 2 is a rational number. A rational number is a number that can be expressed as the quotient of two integers, while an irrational number cannot. Thus, our aim is to make the numerator, which is currently an irrational number, rational.
02

Apply Mathematical Property

We know that the number 1 is neutral for multiplication, and it can be expressed in various forms. In order to make the numerator rational without changing the value of the fraction, multiply it by \(\frac{\sqrt{3}}{\sqrt{3}}\) since \(\frac{\sqrt{3}}{\sqrt{3}}\) is equivalent to 1 and doesn't change the value.
03

Perform the Operation

When we multiply these fractions, we get \(\frac{\sqrt{3} * \sqrt{3}}{2 * \sqrt{3}}\). This simplifies to \(\frac{3}{2 \sqrt{3}}\). In this fraction, the numerator is now rational.
04

Rationalize the Denominator

Although we've achieved a rational numerator, it's common to also have a rational denominator. Thus, multiply again by \(\frac{\sqrt{3}}{\sqrt{3}}\) to get \(\frac{3\sqrt{3}} {6}\), which simplifies to \(\frac{\sqrt{3}} {2}\) which is our final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Rational Numbers
Rational numbers are numbers that can be written as a fraction where both the numerator and the denominator are integers. The denominator should not be zero. For example, \( \frac{1}{2} \) and \( -3 \) are rational numbers.
Rational numbers are easy to work with because they can be exactly expressed as decimals or fractions.
They have either a terminating decimal expansion like 0.75, or a repeating one like 0.333...
  • Examples include all integers, like 5 or -8.
  • Fractions like \( \frac{7}{4} \) also count as rational numbers.
Understanding these characteristics will help you handle mathematical operations more confidently.
Diving into Irrational Numbers
Irrational numbers, unlike rational numbers, cannot be expressed as simple fractions.
They have non-repeating, non-terminating decimal expansions. A perfect example is \(\sqrt{3}\), which is approximately 1.732...
This complexity makes irrational numbers fascinating but also challenging for arithmetic.
  • Numbers like \(\pi\) and \(e\) are well-known examples of irrational numbers.
  • Some square roots, like \(\sqrt{2}\), cannot be simplified into fractions either.
Understanding their properties allows you to rationalize fractions by manipulating terms without breaking mathematical laws.
Multiplying Fractions Effectively
Multiplying fractions involves multiplying their numerators together and then their denominators. It's a straightforward process that simplifies complex expressions.
For example, when multiplying two fractions \( \frac{a}{b} \) and \( \frac{x}{y} \), multiply \( a \times x \) to get the new numerator, and \( b \times y \) for the new denominator.
  • When multiplying fractions, remember to simplify the resulting fraction if possible.
  • Always look out for common factors that can cancel out during multiplication.
This basic rule is essential in rationalizing fractions and simplifying complex expressions.
Simplifying Fractions for Clarity
Simplifying fractions makes them easier to understand and use in calculations.
It involves reducing the fraction to its simplest form, where the numerator and denominator are the smallest integers possible.
To simplify, you find the greatest common divisor (GCD) of the numerator and denominator, and divide them both by it.
  • For example, \( \frac{6}{8} \) simplifies to \( \frac{3}{4} \).
  • Simplifying helps in recognizing equivalent fractions.
This skill is key in ensuring your work is clean and clear, making it easier to communicate mathematical ideas.

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Most popular questions from this chapter

To generate Pythagorean triples, pick natural numbers for \(x\) and \(y(x>y) .\) Let \(a=2 x y\) and \(b=x^{2}-y^{2},\) and \(c=x^{2}+y^{2} .\) Why do you always get a Pythagorean triple?

If \(x>y,\) which is the larger number in each pair? $$\left(\frac{1}{2}\right)^{x} \cdot\left(\frac{1}{2}\right)^{y}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt[3]{1,600 x y^{2} z^{3}}$$

Rationalize the numerator of \(\frac{\sqrt{5}+2}{5}\).

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$a^{3 / 5} a^{-1 / 2}$$
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