Video Transcript
Find the multiplicative inverse of
the square root of 0.36, giving your answer in its simplest form.
In this question, we are asked to
find the multiplicative inverse of the square root of a rational number given as a
decimal. And we need to give our answer in
its simplest form. To do this, we can start by
recalling that the multiplicative inverse of a nonzero, real number 𝑎 is the real
number 𝑏 whose product with 𝑎 is equal to one.
In general, the multiplicative
inverse of a nonzero, real number 𝑎 is one over 𝑎, since 𝑎 times one over 𝑎 is
equal to one. Therefore, the multiplicative
inverse of the square root of 0.36 is one over the square root of 0.36. We need to give our answer in its
simplest form. And there are a few different ways
of doing this. For instance, we could directly
evaluate the square root of 0.36. However, we are going to
rationalize the denominator first. We multiply the numerator and
denominator by the square root of 0.36 to get the square root of 0.36 over 0.36.
To simplify this result further, we
need to rewrite 0.36 as a fraction. We write this as 36 over 100, since
it is the quotient of two perfect squares. Therefore, if we take the square
root of both sides of the equation, we can then take the square root of the
numerator and denominator separately to obtain root 36 over root 100, which is equal
to six over 10. We can then cancel the shared
factor of two in the numerator and denominator to see that the square root of 0.36
is equal to three over five. Therefore, we can rewrite the
numerator as three-fifths to obtain three-fifths divided by 0.36.
We can then note that 0.36 is equal
to 36 over 100. And we can then recall that
dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, instead of dividing by
0.36, we can multiply by 100 over 36. We now need to simplify the
fraction. We can start by canceling the
shared factors of three and two in the numerator and denominator to get 20 over
12. We can then cancel the shared
factor of four in the numerator and denominator to get five-thirds. This cannot be simplified any
further.
Hence, we have shown that the
multiplicative inverse of root 0.36 in its simplest form is five-thirds.
