Express 1400 as a product of its prime factors. Answer ... Hence, explain why 1400 is not a perfect square. Answer ... ... a and b are both prime numbers. Find the value of a and of b such that $1400 \times \frac{a}{b}$ is a perfect cube. Answer a = ... b = ...

1. Divide 1400 repeatedly by prime numbers until all factors are prime.

$1400 = 14 \times 100$

$= (2 \times 7)(2 \times 2 \times 5 \times 5)$

2. Write the prime factors neatly using powers.

$1400 = 2 \times 7 \times 2 \times 2 \times 5 \times 5$

$= 2^3 \times 5^2 \times 7$

1. Use the prime factorisation to check whether all prime powers are even.

From part (a),
$1400=2^3\times5^2\times7^1.$
For a perfect square, every prime factor must have an even power. Here, $2^3$ and $7^1$ have odd powers, so $1400$ is not a perfect square.

1. Use the prime factorisation and identify the exponent changes needed for a perfect cube.

From part (a),
$1400=2^3\times5^2\times7^1.$
For a perfect cube, each power must be a multiple of $3$.

The power of $5$ is $2$, so multiply by $5$ to make it $5^3$.

The power of $7$ is $1$, so divide by $7$ to remove the factor $7$ completely, giving power $0$, which is a multiple of $3$.

2. State the prime values of $a$ and $b$ and check the resulting expression is a perfect cube.

Hence
$a=5,\qquad b=7.$
Check:
$1400\times\frac{5}{7}=2^3\times5^2\times7\times\frac{5}{7}=2^3\times5^3=(10)^3,$
which is a perfect cube.