Question

Solve and find the value of the given expression: ${{2}^{4}}\times {{3}^{2}}\times {{5}^{2}}$

Answer 1

Hint: Power of any number suggests multiplying that number up to the number of values of power times i.e. it indicates how many times base number is going to be multiplied. Use this concept to find the terms in the form of powers and hence solve it.

Complete step by step answer:
Here, we need to determine the value of the expression ${{2}^{4}}\times {{3}^{2}}\times {{5}^{2}}$ .As the numbers ${{2}^{4}},{{3}^{2}}$ and ${{5}^{2}}$ are in multiplication and the numbers involved in the multiplication are in the power form. Hence, we need to simplify the terms of the given expression and hence multiply them to get the value of the whole expression.
So, terms involved with multiplication are ${{2}^{4}},{{3}^{2}}$ and ${{5}^{2}}$.Now, we know the power of any number indicates how many times the base number is going to be multiplied. It means the number of the form ${{a}^{b}}$ can be expressed as multiplication of ‘a’ to ‘b’ times i.e. xaxaxaxaxa ……a (b times). So, representing the product of same number in shorter way or representation can be given with the help of power property i.e. it can be represented as ${{a}^{b}}$ type.
Now we know there are three different numerals represented in powers that are getting multiplies in the given expression. So, values of them with the help of above property can be given as:
First term involved in the expression is ${{2}^{4}}$ .So, we need multiply number ‘2’ by four times as per the rules discussed above. So, we get value of ${{2}^{4}}$ as
${{2}^{4}}=2\times 2\times 2\times 2=16$
Similarly, the second term in the expression is ${{3}^{2}}$.So, we need to multiply number 3 by two times. Hence, we get
${{3}^{2}}=3\times 3=9$
And the third term in the given expression is ${{5}^{2}}$.It means we need to multiply the number 5 to two times. Hence, we can get value of ${{5}^{2}}$as
${{5}^{2}}=5\times 5=25$
Hence, we can get value of the whole expression by multiplying the values of ${{2}^{4}},{{3}^{2}},{{5}^{2}}$as calculated above. Hence, we get
$\begin{align}
  & {{2}^{4}}\times {{3}^{2}}\times {{5}^{2}}=16\times 9\times 25 \\
 & =144\times 25 \\
 & =3600
\end{align}$
So, Value of the given expression in the problem is 3600.
Hence, answer of the given question is 3600

Note: One need to calculate values of ${{2}^{4}},{{3}^{2}},{{5}^{2}}$ individually and then put to the expression. We can calculate value of the given expression in one line as following way:
$\begin{align}
  & {{2}^{4}}\times {{3}^{2}}\times {{5}^{2}}=2\times 2\times 2\times 3\times 3\times 5\times 5 \\
 & =3600
\end{align}$
So, I need not to calculate values individually, just expand them in the expression only. One may confuse the powers of the expression. And an go wrong if he/she write ${{2}^{4}}$as 4 x 4 or ${{3}^{2}}$ as 2 x 2 x2 or ${{5}^{2}}$ as 2 x 2 x 2 x 2 x 2; because we need to multiply the base term to power times not power term to base times. So be clear with the definition and representation of power of numbers.