Exam-Style Problems

Stage 9 - Q1 - 2 marks

18937

Classify as rational or irrational.

\[\sqrt{225}\]

Hint

Check whether the number under the root is a square.

Stage 9 - Q2 - 2 marks

18938

Classify as rational or irrational.

\[\sqrt{290}\]

Hint

Locate the radicand between consecutive squares.

Stage 9 - Q3 - 3 marks

18939

Is this terminating decimal rational?

\[0.43\]

Hint

Write the decimal over a power of ten.

Stage 9 - Q4 - 3 marks

18940

Is this recurring decimal rational?

\[0.\overline{8}\]

Hint

A recurring decimal can be expressed as a fraction.

Stage 9 - Q5 - 4 marks

18941

Classify the product. Explain your answer.

\[\sqrt{531}\times\sqrt{531}\]

Hint

Multiplying a square root by itself reverses the root.

Stage 9 - Q6 - 4 marks

18942

Classify the product. Explain your answer.

\[\sqrt{11}\times\sqrt{11}\]

Hint

Multiplying a square root by itself reverses the root.

Stage 9 - Q7 - 5 marks

18943

Classify the result and justify it.

\[4+\sqrt{2}\]

Hint

Suppose the sum were rational and subtract the integer.

Stage 9 - Q8 - 5 marks

18944

Classify both numbers and explain.

\[\sqrt[3]{-216},\quad \frac{1}{\sqrt{36}}\]

Hint

Simplify each root before deciding.

Stage 9 - Q9 - 5 marks

18945

A calculator displays 8.414213562 for a number. Does that display prove that the exact number is rational?

Hint

Separate an exact decimal from a rounded display.

Stage 9 - Q10 - 3 marks

18946

Find two different irrational numbers whose sum and product are both rational. Show why they work.

Hint

Try an irrational number and its additive inverse.