Exam-Style Problems

Stage 9 - Q1 - 2 marks

19037

Simplify.

\[2x^{2}\times4x^{4}\]

Hint

Treat the numbers and powers separately.

Stage 9 - Q2 - 2 marks

19038

Simplify for nonzero \(x\).

\[\frac{16x^{8}}{4x^{4}}\]

Hint

The denominator must be nonzero.

Stage 9 - Q3 - 3 marks

19039

Simplify.

\[(-6x^{6})(5x^{1})\]

Hint

Multiply the numerical coefficients, then combine the powers.

Stage 9 - Q4 - 3 marks

19040

Simplify.

\[8x^{3}y^2\times5xy^{2}\]

Hint

Keep the \(x\) indices and \(y\) indices separate.

Stage 9 - Q5 - 4 marks

19041

Simplify for nonzero \(x\).

\[\frac{3x^{7}}{6x^{2}}\]

Hint

A non-integer coefficient is an exact fraction.

Stage 9 - Q6 - 4 marks

19042

Simplify for nonzero \(x\).

\[\frac{5x^{5}}{6x^{3}}\]

Hint

A non-integer coefficient is an exact fraction.

Stage 9 - Q7 - 5 marks

19043

Find \(n\) so the identity holds for every nonzero \(x\).

\[x^n\times x^{4}=x^{8}\]

Hint

This is an identity in \(x\), not an equation for one special \(x\).

Stage 9 - Q8 - 5 marks

19044

Explain why these expressions are different.

\[2x^{6}+7x^{6},\quad(2x^{6})(7x^{6})\]

Hint

State which operation is being performed.

Stage 9 - Q9 - 5 marks

19045

Simplify, then evaluate at \(x=2\). Assume \(x\) is nonzero.

\[\frac{8x^{4}}{2x^{1}}\]

Hint

Cancelling first reduces the arithmetic.

Stage 9 - Q10 - 3 marks

19046

Positive integers \(p\) and \(q\) satisfy both identities for every nonzero \(x\). Find \(p\) and \(q\).

\[x^p x^q=x^{12},\quad\frac{x^p}{x^q}=x^4\]

Hint

Translate the product and quotient into two statements about indices.