Question

How do you simplify ${5^2} \times {5^4} = {5^{2 + 4}}$${5^2} \times {5^4}?$

Answer 1

Hint: Here we need to simplify the given problem using some basic algebraic expressions. In the above problem we have given a number which is raised to some power. Since the base numbers are the same i.e. 5, we use indice law to simplify it. The Indice law which we use here is given by ${x^n} \times {x^m} = {x^{n + m}}$. After applying this law we simplify the problem further. We then use the exponential rule to obtain the desired solution.

Complete step by step solution:
In the given problem we need to simplify ${5^2} \times {5^4}$.
Here first we note that the base number for the above problem is the same, which is 5. Also the base number 5 is raised to some power. So we make use of indice law to simplify it.
According to indice law, we need to add the powers of the number due to the product of two exponentials.
i.e. ${x^n} \times {x^m} = {x^{n + m}}$ ……(1)
In the above problem we have $x = 5$,$n = 2$ and $m = 4$.
Substituting the values of x, n, m in the equation (1), we get,
${5^2} \times {5^4} = {5^{2 + 4}}$ ……(2)
Now we add the powers of 5 in the right hand side.
We know that $2 + 4 = 6$.
Hence substituting this in the equation (2), we get,
$ \Rightarrow {5^2} \times {5^4} = {5^6}$.
Now we try to get the simplified form of ${5^6}$.
We have the exponential rule which is given by,
    $\underbrace {a \times a \times a \times ....}_{k - times} = {a^k}$ ……(3)
Here we have the number 5 raised to the power 6. So, here the value of k is, $k = 6$.
Now we make use of the exponential rule to simplify ${5^6}$. So we need to multiply the number 5 six times to get the solution.
By the equation (3) for $k = 6$, we have,
$5 \times 5 \times 5 \times 5 \times 5 \times 5 = {5^6}$ ……(4)
We know that,
 $5 \times 5 = 25$
$25 \times 5 = 125$
$125 \times 5 = 625$
$625 \times 5 = 3125$
$3125 \times 5 = 15625$
Hence in the equation (4), we get,
$ \Rightarrow $${5^6} = 15625$.

Therefore ${5^2} \times {5^4} = {5^6} = 15625$.

Note :
Alternative method :
It is easier to think about the above problem by writing it out like this which is given below.
We need to simplify ${5^2} \times {5^4}$.
Firstly write ${5^2}$ as $5 \times 5$.
Also write ${5^4}$as $5 \times 5 \times 5 \times 5$.
So, ${5^2} \times {5^4}$ becomes,
${5^2} \times {5^4} = \left( {5 \times 5} \right) \times \left( {5 \times 5 \times 5 \times 5} \right)$
Now count the number of five’s on the R.H.S., which is equal to 6.
Hence, ${5^2} \times {5^4} = {5^6}$.
This is easier to think and solve.
We can further simplify ${5^6}$ as explained in the main solution which is given above the note.