If3+2root 3/3-root3=a+root3
, then the value of a+rootb where a and b are rational numbers is
Answers
Answer:
Answer:
\bigstar{\bold{Value\:of\:a=\dfrac{5}{2} }}★Valueofa=
2
5
\bigstar{\bold{Value\:of\:b=\dfrac{3}{2} }}★Valueofb=
2
3
Step-by-step explanation:
\Large{\underline{\underline{\bf{Given:}}}}
Given:
\sf{\dfrac{3+2\sqrt{3} }{3-\sqrt{3} }=a+b\sqrt{3} }
3−
3
3+2
3
=a+b
3
\Large{\underline{\underline{\bf{To\:Find:}}}}
ToFind:
The values of a and b
\Large{\underline{\underline{\bf{Solution:}}}}
Solution:
→ Given,
\sf{\dfrac{3+2\sqrt{3} }{3-\sqrt{3} } }
3−
3
3+2
3
→ Rationalising the denominator by multiplying 3 + √3 on both numerator and denominator
\sf{\dfrac{3+2\sqrt{3}\times (3+\sqrt{3}) }{3-\sqrt{3}\times (3+\sqrt{3} ) } }
3−
3
×(3+
3
)
3+2
3
×(3+
3
)
→ Simplifying by using identities,
\sf{\dfrac{9+3\sqrt{3}+6\sqrt{3} +6 }{3^{2} -(\sqrt{3})^{2} } }
3
2
−(
3
)
2
9+3
3
+6
3
+6
\sf{=\dfrac{9+9\sqrt{3}+6 }{9-3} }=
9−3
9+9
3
+6
→ Taking 3 common from the numerator,
\sf{=\dfrac{3(3+3\sqrt{3}+2) }{6} }=
6
3(3+3
3
+2)
→ Cancelling the numerator and denominator by 3
\sf{=\dfrac{(3+3\sqrt{3}+2) }{2} }=
2
(3+3
3
+2)
\sf{=\dfrac{3}{2}+\dfrac{3\sqrt{3} }{2}+\dfrac{2}{2} }=
2
3
+
2
3
3
+
2
2
= 5/2 + 3/2 ×√3
→ Equating it we get the value of a as 5/2 and b as 3/2
\boxed{\bold{Value\:of\:a=\dfrac{5}{2} }}
Valueofa=
2
5
\boxed{\bold{Value\:of\:b=\dfrac{3}{2} }}
Valueofb=
2
3
\Large{\underline{\underline{\bf{Notes:}}}}
Notes:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b²= (a + b) (a - b)
Answer:
Given, 3+2root 3/3-root3
∵Rationalising the denominator by multiplying 3 + √3 on both numerator and denominator
=> 3+2root 3 × (3+ root 3) / 3-root 3 × 3+root 3
∵Simplifying by using identities,
=> 9+3root 3 + 6root 3 +6 / 3² - (root 3)²
=> 9 + 9root 3 + 6 / 9 - 3
∵Taking 3 common from the numerator,
=> 3(3 + root 3 + 2) / 6
∵Cancelling the numerator and denominator by 3,
=> 3 + 3root 3 + 2 / 2
=> 3/2 + 3root 3/2 + 2/2
=> 5/2 + 3/2 × root 3
∵Equating it we get the value of a as 5/2 and b as 3/2,
Value of a = 5/2.
Value of b = 3/2.