3 (a) Simplify, giving your answer as a single power of 7.

(i) \(7^{5} \times 7^{6}\)
............................................. [1]

(ii) \(7^{15} \div 7^{5}\)
............................................. [1]

(iii) \(42+7\)
............................................. [1]

(b) Simplify.
\[
\left(5 x^{2} \times 2 x y^{4}\right)^{3}
\]
............................................. [3]

(i) Find the highest common factor (HCF) of \(P\) and \(Q\).
............................................. [2]

(ii) Find the lowest common multiple (LCM) of \(P\) and \(Q\).
............................................. [2]

(iii) \(P \times R\) is a cube number, where \(R\) is an integer.
Find the smallest possible value of \(R\).
............................................. [2]

(d) Factorise the following completely.

(i) \(x^{2}-3 x-28\)
............................................. [2]

(ii) \(7(a+2 b)^{2}+4 a(a+2 b)\)
............................................. [2]

(e) \(\quad 3^{2 x-1}=\frac{1}{9^{x}} \times 3^{2 y-x}\)
Find an expression for \(y\) in terms of \(x\).
\[
y=...............................
\] [4]

Simplify, giving your answer as a single power of 7. $$\(42+7\)$$ .............................................

Answer:

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