#5sqrt5# can be written in the form #5^k#. What is the value of #k#?

1 Answer

#5sqrt5=5xxsqrt5=5^1xx5^(1/2)=5^(1+1/2)=5^(3/2) :. k=3/2#

Explanation:

Let's first break this down:

#5sqrt5=5xxsqrt5#

Now let's talk exponents.

Exponents tell us the number of times we multiply something by. For instance, if we have #3^2#, we know we multiply 3 by itself twice, which is #3xx3# and that gives us 9.

So how many times do we multiply 5 by to get 5? Answer: once. Therefore:

#5=5^1#

Now let's talk about the square root. If we take #sqrt5# and multiply by itself twice, we get:

#sqrt5xxsqrt5=5#

Remember that when we are multiplying numbers with exponents and the same base, the rule is:

#x^axxx^b=x^(a+b)#

and so we can see that #sqrt5=5^(1/2)#:

#sqrt5xxsqrt5=5^(1/2)xx5^(1/2)=5^(1/2+1/2)=5^1#

Which gives us:

#5sqrt5=5xxsqrt5=5^1xx5^(1/2)=5^(1+1/2)=5^(3/2)#

Therefore, #k=3/2#