Problem 13 Give an example of two irrationa... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 0: Problem 13

Give an example of two irrational numbers whose product is a rational number.

Short Answer

Expert verified

The two irrational numbers whose product is a rational number are $\sqrt{2}$ and $\dfrac{1}{\sqrt{6}}$. Their product is $\dfrac{1}{3}$, which is a rational number.

Step by step solution

Recall the Properties of Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction (i.e., as the quotient of two integers). Examples include square roots of non-perfect square numbers, 蟺 (pi), and e (Euler's number).

Choose an Irrational Number

Let's start by choosing a well-known irrational number. We'll use the square root of 2 ($\sqrt{2}$), which is a known irrational number.

Create Another Irrational Number from the Chosen Number

Now, we'll create another irrational number using the chosen number. Let's multiply $\sqrt{2}$ by another irrational number, say $\sqrt{3}$. The product of these two irrational numbers will also be an irrational number. So we have our second irrational number: $\sqrt{2} \cdot \sqrt{3} = \sqrt{6}$.

Create a Product of the Two Irrational Numbers that is a Rational Number

Now let's find the product of our two irrational numbers ($\sqrt{2}$ and $\sqrt{6}$) in such a way that it results in a rational number. To do this, we can multiply the first irrational number by the reciprocal of the second irrational number. The reciprocal of $\sqrt{6}$ is $\dfrac{1}{\sqrt{6}}$. Now, let's multiply $\sqrt{2}$ by the reciprocal of $\sqrt{6}$: $\sqrt{2} \cdot \dfrac{1}{\sqrt{6}}$

Simplify the Product

To simplify the product, let's rationalize the denominator by multiplying the numerator and denominator by $\sqrt{6}$: $\sqrt{2} \cdot \dfrac{1}{\sqrt{6}} = \dfrac{\sqrt{2} \cdot \sqrt{6}}{6}$ This simplifies to: $\dfrac{\sqrt{12}}{6}$

Confirm the Product is Rational

We can further simplify the product: $\dfrac{\sqrt{12}}{6} = \dfrac{2\sqrt{3}}{6} = \dfrac{1}{3}$ The product of our two irrational numbers ($\sqrt{2}$ and $\dfrac{1}{\sqrt{6}}$) is $\dfrac{1}{3}$, which is a rational number. Thus, we have found two irrational numbers whose product is a rational number: $\sqrt{2}$ and $\dfrac{1}{\sqrt{6}}$

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91影视

Give an example of two irrational numbers whose product is a rational number.

Short Answer

Step by step solution

Recall the Properties of Irrational Numbers

Choose an Irrational Number

Create Another Irrational Number from the Chosen Number

Create a Product of the Two Irrational Numbers that is a Rational Number

Simplify the Product

Confirm the Product is Rational

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