Question 14 of 26intermediate🔧 ApplyLong Answer3 marks

Let xx and yy be rational numbers. Show that xyxy is a rational number.

Correct Answer

Let xx and yy be two rational numbers. By definition, we can write

x=mnx = \frac{m}{n}

and

y=pqy = \frac{p}{q}

where m,n,p,qm, n, p, q are integers and n≠0,q≠0n \neq 0, q \neq 0.

We calculate the product as

xy=mn×pq=mpnqxy = \frac{m}{n} \times \frac{p}{q} = \frac{mp}{nq}

Using the properties of integers, we see that mpmp and nqnq are integers.

Since n≠0n \neq 0 and q≠0q \neq 0, it follows that nq≠0nq \neq 0. Therefore,

xy=mpnqxy = \frac{mp}{nq}

is a rational number.

Exercise: EXERCISE A1.3 | Q: 4 | (Page 11)
For More Understanding

Explanation

The provided context outlines the proof for the sum of two rational numbers, establishing that rational numbers are ratios of integers and that the product of denominators is non-zero. Applying this same logic to the product xyxy, we represent xx and yy as fractions of integers.

The context explicitly notes that 'Using the properties of integers... nqnq are integers', confirming that the product of integers is an integer. Thus, the numerator mpmp and denominator nqnq are integers, and nq≠0nq \neq 0, satisfying the definition of a rational number.

Solution Steps

  1. Step 1: Define xx and yy as ratios of integers (x=mn,y=pqx = \frac{m}{n}, y = \frac{p}{q}).

  2. Step 2: Express the product xyxy as a single fraction (mpnq\frac{mp}{nq}).

  3. Step 3: State that mpmp and nqnq are integers using properties of integers.

  4. Step 4: Conclude that xyxy is rational since the denominator nq≠0nq \neq 0.

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