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HomeClass 10mathematicsChapter 1Exercise 1.2Q2
| Exercise 1.2

Question 2

Prove that 32+53\sqrt{2} + 5 is irrational.

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SOLUTION

UNDERSTAND THE QUESTION

The problem asks to show that the number 32+53\sqrt{2}+5 cannot be expressed as a ratio of two integers. The core idea is a proof by contradiction using the known fact that 2\sqrt{2} is irrational. If 32+53\sqrt{2}+5 were rational, simple algebraic manipulation would imply 2\sqrt{2} is rational, contradicting the established result.

STEP 1 — ASSUME THE CONTRARY

Suppose, for the sake of contradiction, that 32+53\sqrt{2}+5 is a rational number. Then there exist integers pp and q≠0q\neq0 such that

32+5=pq.3\sqrt{2}+5=\dfrac{p}{q}.

STEP 2 — ISOLATE 2\sqrt{2}

Subtract 55 from both sides and then divide by 33:

2=pq−53=p−5q3q.\sqrt{2}=\dfrac{\dfrac{p}{q}-5}{3}=\dfrac{p-5q}{3q}.

The right‑hand side is a ratio of two integers, hence rational.

STEP 3 — RECALL THE IRRATIONALITY OF 2\sqrt{2}

It is a classic result that 2\sqrt{2} is irrational. A standard proof assumes 2=ab\sqrt{2}=\dfrac{a}{b} in lowest terms, squares both sides to obtain 2b2=a22b^{2}=a^{2}, and deduces that both aa and bb must be even, contradicting the assumption that the fraction is reduced.

STEP 4 — DERIVE THE CONTRADICTION

Step 2 shows 2\sqrt{2} would be rational, while Step 3 states 2\sqrt{2} is irrational. This contradiction means our original assumption is false.

STEP 5 — CONCLUDE

Therefore 32+53\sqrt{2}+5 cannot be rational; it is irrational.

ANSWER

32+53\sqrt{2}+5 is irrational.

COMMON MISTAKES

Assuming that adding a rational number (like 55) to an irrational automatically yields an irrational without justification; forgetting to isolate 2\sqrt{2} correctly; overlooking the need to prove 2\sqrt{2} is irrational as a separate lemma.

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