Show that seven divided by three
minus root two can be rewritten in the form 𝑎 plus 𝑏 root two, where 𝑎 and 𝑏 are
In order to simplify or rewrite
this expression, we need to multiply the top and the bottom by the conjugate of the
bottom. The conjugate of three minus root
two is three plus root two. The two terms remain the same. But the sign changes, either from
positive to negative or in this case from negative to positive. The conjugate of 𝑎 minus root 𝑏
is 𝑎 plus root 𝑏.
Our next step is to multiply the
two numerators and then separately multiply the two denominators. In order to multiply the numerator,
we multiply seven by three and seven by root two. Seven multiplied by three is
21. And seven multiplied by root two is
seven root two.
In order to expand the two terms on
the bottom, we’ll use the FOIL method. Multiplying the first terms gives
us nine. Multiplying the outside terms gives
us three root two. Multiplying the inside terms gives
us negative three root two. And finally, multiplying the last
terms gives us negative two. This is because root two multiplied
by root two is equal to root four. And the square root of four is
equal to two.
As root two multiplied by root two
is equal to two, then negative root two times root two is equal to negative two. We can then cancel the middle
terms, as three root two minus three root two is zero.
Finally, nine minus two is equal to
seven. Therefore, our dominator is
seven. Seven divided by three minus root
two can be rewritten as 21 plus seven root two divided by seven.
Our final step is to cancel by
dividing each of the three terms by seven. This gives us a final answer of
three plus root two. We’ve rewritten seven divided by
three minus root two in the form 𝑎 plus 𝑏 root two, where 𝑎 equals three and 𝑏