Question

Write 243 as a product of prime factors and express it in exponential form.

Answer 1

Hint: We solve this question by using the prime factorization method to obtain the prime factors of the number given which is 243. We try dividing by the prime numbers until we get a quotient of 1. Then we take the product of the prime factors and represent multiples of them in terms of exponentials by raising it to the power of as many numbers as there are.

Complete step-by-step solution:
In order to solve this question, let us first find out the prime factors of the given number 243. This can be done by using the prime factorization method. This method specifies that we go on dividing the given number by the prime numbers such that we get a final value of 1. Let us apply this prime factorization method for 243.
We represent this as,
$\Rightarrow |\underset{―}{243}$
Now, we try dividing this by the smallest prime number first which is 2. We cannot divide it since it 243 is not a multiple of 2. Next, we need to try to divide by 3. Since 243 is a multiple of 3, we can divide the above number by 3.
$\begin{array}{rl}& \Rightarrow 3|\underset{―}{243}\\ & |\underset{―}{81}\end{array}$
Next, we again divide 81 by 3 since it cannot be divided by 2 and it is a multiple of 3.
$\begin{array}{rl}& \Rightarrow 3|\underset{―}{243}\\ & \text{ 3}|\underset{―}{81}\\ & |\underset{―}{27}\end{array}$
Similarly, we keep on dividing by 3 since it is a multiple of 3 until we get a 1.
$\begin{array}{rl}& \Rightarrow 3|\underset{―}{243}\\ & \text{ 3}|\underset{―}{81}\\ & \text{ 3}|\underset{―}{27}\\ & \text{ 3}|\underset{―}{9}\\ & \text{ 3}|\underset{―}{3}\\ & 1\end{array}$
Therefore, we have factored it using the prime factorization method. Next, we need to represent 243 as a product of these prime factors as,
$\Rightarrow 243=3\times 3\times 3\times 3\times 3$
Since it is a product of 5 consecutive 3s, this can be represented in exponential form as,
$\Rightarrow 243=3^5$
Hence, we have represented 243 as a product of prime given as $243=3\times 3\times 3\times 3\times 3,$ and in the exponential form as $243=3^5.$

Note: We need to be careful while applying the prime factorization method. We need to try to divide the number always starting from the smallest prime number which is 2. If it is not divisible, then we need to go for the next prime number which is 3. If it is not divisible by 3 also then we need to go on to the next prime number which is 5 and so on.