A conjugate and a rationality claim

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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Before you start

Surds and exact answers. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

Let u = 6 + √35 and v = 6 − √35.

Part a

Show that uv is rational, and state its value.

Part b

Hence express 1/u without a denominator.

Part c

A student says that a rational product means both factors are rational. Give a counterargument.

Model solution and review criteria

Part a

uv = 36 − 35 = 1. The mixed terms cancel.

Multiply conjugates and use (√b)² = b.
Show the rational value, not merely a decimal.

Part b

Since uv = 1, 1/u = v = 6 − √35.

Use the earlier product.
State the exact expression.

Part c

35 lies strictly between (5)² and $6^{2}$ , so its square root is irrational. Both factors here are irrational despite their product being 1.

Explain why b is not a perfect square.
Use these factors as a counterexample.

Compare your own reasoning. No automatic examination marks are awarded.