Chapter 4.1, Problem 67E

To determine

To write: The prime factorizations, with exponents, for the perfect squares of $4,9,16,25,36\text{ and }64$ . Also make a conjecture about the exponents in the prime factorisation of a perfect square.

Answer to Problem 67E

Prime factorisation of the given numbers are

  $4=2^2,9=3^2,16=2^4,25=5^2,36=2^2\times {\text{3}}^2\text{ and }64=2^6$

Explanation of Solution

Given information:

The square of an integer is called a perfect square.

Given numbers are $4,9,16,25,36\text{ and }64$

Consider the numbers as $4,9,16,25,36\text{ and }64$ ,

Now,

  $\begin{array}{c}4=2\times 2=2^2\\ 9=3\times 3=3^2\\ 16=4\times 4=2\times 2\times 2\times 2=2^4\\ 25=5\times 5=5^2\\ 36=6\times 6=3\times 2\times 3\times 2=2^2\times 3^2\\ 64=8\times 8=2\times 2\times 2\times 2\times 2\times 2=2^6\end{array}$

Therefore, prime factorisation of the given numbers are

  $4=2^2,9=3^2,16=2^4,25=5^2,36=2^2\times {\text{3}}^2\text{ and }64=2^6$