Calculate the square root of one
and seven-ninths minus 19 over 32.
In this question, we are asked to
evaluate an expression involving the square root of a mixed number and then the
difference of this result with a rational number.
To do this, we first recall that
the square root of a number 𝑎 is the nonnegative number whose square is 𝑎. It is easier to find the square
root of a fraction than a mixed number. So we will convert one and
seven-ninths into a fraction by rewriting one as nine over nine. We obtain the square root of 16
over nine minus 19 over 32.
Now that we have the square root of
a fraction, we can recall that quotient rule for square roots tells us that if 𝑎
and 𝑏 are integers with 𝑎 nonnegative and 𝑏 positive, then the square root of 𝑎
over 𝑏 is equal to the square root of 𝑎 over the square root of 𝑏. Applying this result with 𝑎 equal
to 16 and 𝑏 equal to nine gives us the square root of 16 over the square root of
nine minus 19 over 32.
We can now evaluate both of the
square roots by recalling that if 𝑐 is a nonnegative number, then the square root
of 𝑐 squared is just equal to 𝑐. Therefore, since 16 is equal to
four squared and nine is equal to three squared, we can evaluate the square roots to
get four-thirds minus 19 over 32.
We now need to find the difference
between the two fractions. And to do this, we need them to
have the same denominator. We can find that the lowest common
multiple of the denominators is 96. So we will multiply the numerator
and denominator of the first fraction by 32 and the numerator and denominator of the
second fraction by three. This gives us 128 over 96 minus 57
over 96, which we can calculate is equal to 71 over 96. We cannot simplify this fraction
any further, so our final answer is 71 over 96.