Question

If 7+4√3 / 7 - 4√3 =X +Y√3 then X = ( )

Answer 1

Solving for 2(−14)2
x
2
(
x
−
14
)
2
for =7+43√√7−43√√
x
=
7
+
4
3
7
−
4
3


Like he said, we should rationalize the denominator using the identity (+)(−)=2−2
(
a
+
b
)
(
a
−
b
)
=
a
2
−
b
2
This identity is called the difference of squares identity

and the goal of rationalization is to multiply the top and bottom by a number such that it will get rid of the radicals in the denominator - this number is called the conjugate

for example the conjugate of ‾√+√
a
+
b
is ‾√−√
a
−
b
as multiplying the two together will get rid of the radicals:

(‾√+√)(‾√−√)=−
(
a
+
b
)
(
a
−
b
)
=
a
−
b
(from the difference of squares identity)

Let’s apply this to the question :

=7+43√√7−43√√
x
=
7
+
4
3
7
−
4
3


The conjugate of the denominator 7−43‾√‾‾‾‾‾‾‾‾√
7
−
4
3
is 7+43‾√‾‾‾‾‾‾‾‾√
7
+
4
3


Multiply the entire expression by the conjugate to get:

7+43√√7−43√√⋅7+43√√7+43√√
7
+
4
3
7
−
4
3
⋅
7
+
4
3
7
+
4
3


=(7+43√)2√(72−(43√)2)√
=
(
7
+
4
3
)
2
(
7
2
−
(
4
3
)
2
)


=7+43√49−16(3)√
=
7
+
4
3
49
−
16
(
3
)


=7+43√1√
x
=
7
+
4
3
1


=7+43‾√
x
=
7
+
4
3


also notice that −14=7+43‾√−14=43‾√−7
x
−
14
=
7
+
4
3
−
14
=
4
3
−
7


2(−14)2=((−14))2
x
2
(
x
−
14
)
2
=
(
x
(
x
−
14
)
)
2


=((43‾√+7)(43‾√−7))2
=
(
(
4
3
+
7
)
(
4
3
−
7
)
)
2


Use the difference of squares identity again with =43‾√
a
=
4
3
and =7
b
=
7
to get

((43‾√+7)(43‾√−7))2=(48−49)2
(
(
4
3
+
7
)
(
4
3
−
7
)
)
2
=
(
48
−
49
)
2


=(−1)2=1
=
(
−
1
)
2
=
1


∴2(−14)2=1
∴
x
2
(
x
−
14
)
2
=
1

Answer 2

Answer:

Let’s apply this to the question :

x=7+43√√7−43√√

The conjugate of the denominator 7−43–√−−−−−−−√ is 7+43–√−−−−−−−√

Multiply the entire expression by the conjugate to get:

7+43√√7−43√√⋅7+43√√7+43√√

=(7+43√)2√(72−(43√)2)√

=7+43√49−16(3)√

x=7+43√1√

x=7+43–√

also notice that x−14=7+43–√−14=43–√−7

x2(x−14)2=(x(x−14))2

=((43–√+7)(43–√−7))2

Use the difference of squares identity again with a=43–√ and b=7 to get

((43–√+7)(43–√−7))2=(48−49)2

=(−1)2=1

∴x2(x−14)2=1