How do you multiply expressions written in scientific notation?

Expressions can be easily multiplied when written in scientific notation by:
1. First, multiplying the numbers other than the powers of 10.
2. Second, multiplying the powers of 10
And then, writing them as a product.

Let us take the general case first.

Multiplying two numbers #x*10^m# and #y*10^n#

First, multiplying the numbers other than the powers of 10, we get:
#x*y=xy#

Second, multiplying the powers of 10 we get
#10^m*10^n=10^(m+n)#

And then writing them as a product, we get
#xy*10^(m+n)#

Therefore, #(x*10^m)*(y*10^n)=xy*10^(m+n)#

Note: When the bases of 2 numbers are equal, their powers can be added up!
Examples:
1). #2^a*2^b=2^(a+b)#
2) #3^3*3^7=3^(3+7)=3^10#

Now, let's take some specific examples.

Q: Multiply #1.2*10^3# and #2.3*10^4#

A:

#(1.2*10^3)*(2.3*10^4)#
#=(1.2*2.3)*(10^(3+4))#
#=2.76*10^7#

Q: Multiply #9.32*10^21# and #8.21*10^32#

A:

#(9.32*10^21)*(8.21*10^32)#
#=(9.32*8.21)*(10^(21+32))#
#=76.5172*10^53#

Notice that this answer is not in the standard form. So, converting this into standard form, we get:

#=7.65172*10^54#