Express 13 824 as a product of its prime factors. Answer ... Explain why 13 824 is a perfect cube. Answer ... ... Given that a is a prime number, find the value of a such that $\frac{1}{8a} \times 13\,824$ is a perfect square. Answer a = ...

1. Divide repeatedly by prime factors until only primes remain.

$13\,824=2\times 6912$
$=2^2\times 3456$
$=2^3\times 1728$
$=2^4\times 864$
$=2^5\times 432$
$=2^6\times 216$
$=2^7\times 108$
$=2^8\times 54$
$=2^9\times 27$
$=2^9\times 3^3$

1. Use the prime factorisation to show all prime powers can form a cube.

From part (a),
$13\,824=2^9\times 3^3.$
Since the exponents $9$ and $3$ are both multiples of $3$,
$2^9\times 3^3=(2^3\times 3)^3=24^3.$
Therefore $13\,824$ is a perfect cube.

1. Substitute the prime factorisation and simplify the expression.

From part (a),
$13\,824=2^9\times 3^3.$
So
$\frac{1}{8a}\times 13\,824=\frac{2^9\times 3^3}{2^3a}=\frac{2^6\times 3^3}{a}.$

2. Choose the prime value of $a$ so that all powers are even.

For a perfect square, all prime powers must have even indices.

$2^6\times 3^3\div a$

Since $a$ is prime, choose $a=3$ so that
$\frac{2^6\times 3^3}{3}=2^6\times 3^2,$
which is a perfect square.