define rationalization,rationalising factor and simplify the following by rationalizing the denominator 2√3-√5/2√2-3√3
Answers
Step-by-step explanation:
We rationalise the denominator of (
3
2
−
5
5
2
−
3
) by multiplying its rationalising factor (3
2
+
5
) in the numerator and denominator as follows:
3
2
−
5
5
2
−
3
×
3
2
+
5
3
2
+
5
=
(3
2
−
5
)(3
2
+
5
)
(5
2
−
3
)(3
2
+
5
)
=
(3
2
)
2
−(
5
)
2
5
2
(3
2
+
5
)−
3
(3
2
+
5
)
=
(9×2)−5
(5×3
2×2
)+(5
2×5
)−(3
3×2
)−(
3×5
)
=
18−5
15
4
+5
10
−3
6
−
15
=
13
(15×2)+5
10
−3
6
−
15
=
13
30+5
10
−3
6
−
15
Hence,
3
2
−
5
5
2
−
3
=
13
30+5
10
−3
6
−
15
For example, the rationalizing factor of √5 is √5 and rationalizing factor of ∛2 is ∛2^2 or ∛4. Since, √5 × √5 = 5 and ∛2 × ∛2^2 = ∛(2 × 2^2) = ∛2^3 = 2
2. (a√z) × (b√z) = (a × b) × (√z × √z) = ab(√z)^2 = abz, which is rational. Therefore, each of the surds a√z and b√z is a rationalizing factor of the other.
3. √5 × 2√5 = 2 × (√5)^2 = 3 × 5 = 15, which is rational. Therefore, each of the surds √5 and 2√5 is a rationalizing factor of the other.
4. (√a + √b) × (√a - √b) = (√a)^2 - (√b)^2 = a - b, which is rational. Therefore, each of the surds (√a + √b) and (√a - √b) is a rationalizing factor of the other.
5. (x√a + y√b) × (x√a - y√b) = (x√a)^2 - (y√b)^2 = ax - by, which is rational. Therefore, each of the surds (x√a + y√b) and (x√a - y√b) is a rationalizing factor of the other.
6. (4√7 + √3) × (4√7 - √3) = (4√7)^2 - (√3)^2 = 112 - 3 = 109, which is rational. Therefore, each of the surd factors (4√7 + √3) and (4√7 - √3) is a rationalizing factor of the other.
7. Also rationalizing factor of ∛(ab^2c^2) is ∛(a^2bc) because ∛(ab^2c^2) × ∛(a^2bc) = abc.
8. Rationalize the following surds.
3√22√2, 63√, 5√33√3
Solution:
3√22√2
= 3√×2√2√×2√
= 6√2.
63√2
= 6×3√3√×3√
= 63√3
= 23–√.
5√33√3
Here the denominator is 3–√3 or 313, for this surd of order 3, rationalization factor will be 323.
= 5√3×323313×323
= 5√3×32√33
= 45√33.
9. Find the rationalizing factor of (√x - ∛y).
Solution:
Let, √x = x^1/2 = a and ∛y = y^1/3 = b.
Now, the order of the surds √x and ∛y and 3 respectively and the L.C.M. of 2 and 3 is 6.
Therefore,
a^6 = (x^1/2)^6 = x^3 and b^6 = (y^1/3)^6 = y^2.
Therefore, a^6 and b^6 both are rational and as such (a^6 - b^6) is also rational.
Now, a^6 - b^6 = (a - b)(a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4 + b^5)
Therefore the rationalizing factor of (a - b) = (√x - ∛y) is (a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4 + b^5) = x^5/2 + x^2y^1/3 + x^3/2y^2/3 + xy + x^1/2y^4/3 + y^5/3
10. Rationalize the surd 2√2+12√2−1.
Step-by-step explanation:
will discuss about the rationalization of surds. When the denominator of an expression is a surd which can be reduced to an expression with rational denominator, this process is known as rationalizing the denominator of the surd.
If a surd or surd with rational numbers present in the denominator of an equation, to simplify it or to omit the surds from the denominator, rationalization of surds is used. Surds are irrational numbers but if multiply a surd with a suitable factor, result of multiplication will be rational number. This is the basic principle involved in rationalization of surds. The factor of multiplication by which rationalization is done, is called as rationalizing factor. If the product of two surds is a rational number, then each surd is a rationalizing factor to other. Like if 2–√ is multiplied with 2–√, it will 2, which is rational number, so 2–√ is rationalizing factor of 2–√.
In other words, the process of reducing a given surd to a rational form after multiplying it by a suitable surd is known as rationalization.
When the product of two surds is a rational number, then each of the two surds is called rationalizing factor of the other.
For example, 5√2√ is a surd where 5–√ is numerator and 2–√ is denominator. Now for rationalization of surds, if we multiply both numerator and denominator by 2–√, then denominator will be a rational number.
5√2√ = 5√×2√2√×2√
= 10√2.