Video Transcript
Given that negative 47 over root
two minus seven is equal to 𝑎 root two plus 𝑏, find the values of 𝑎 and 𝑏.
To rewrite this quotient in the
requested form, we need to rationalize its denominator. Currently, the value of root two in
the denominator means that it is irrational. To rationalize the denominator of a
quotient, we multiply both the numerator and denominator by a value that is
carefully chosen because its product with the denominator is rational. Because we multiply both the
numerator and denominator by the same value, we’re multiplying by one overall, and
so the result is equivalent to the original fraction.
We first recall that the conjugate
of the radical expression 𝑝 plus 𝑞 root 𝑟, where 𝑝, 𝑞, and 𝑟 are rational
numbers and 𝑟 is nonnegative, is equal to 𝑝 minus 𝑞 root 𝑟. Secondly, the product of such an
expression and its conjugate is always rational. 𝑝 plus 𝑞 root 𝑟 multiplied by 𝑝
minus 𝑞 root 𝑟 is equal to 𝑝 squared minus 𝑞 squared 𝑟. If we rewrite the denominator of
root two minus seven so that it exactly matches this form, we have negative seven
plus root two. And hence, 𝑝 is equal to negative
seven, 𝑞 is equal to one, and 𝑟 is equal to two. The conjugate of this value is
negative seven minus root two.
Multiplying the fractions and
applying the quoted result gives negative 47 multiplied by negative seven minus root
two in the numerator and negative seven squared minus one squared multiplied by two
in the denominator. The denominator simplifies to 49
minus two, which is 47. We can then cancel the shared
factor of 47 in the numerator and denominator. Finally, distributing the negative
one over the parentheses in the numerator gives seven plus root two. Reordering the terms and then
comparing this to the required form, we find that the value of 𝑎 is one and the
value of 𝑏 is seven.
So, by rationalizing the
denominator of the given quotient, we found that it is equivalent to root two plus
seven. And hence, 𝑎 is equal to one and
𝑏 is equal to seven.
