\begin{align*} \text {We know that} \: 3 \times 3 &= 9 \\ or, \: 3^2 &= 9 \\ or, \: 3^{2 \times \frac {1}{2}} &= 9^{\frac{1}{2}} \\ or, \: 3 &=9^{\frac{1}{2}} \\ or, \: 3 &= \sqrt{9} \\ \end{align*}
The square root of 9 is 3.
\begin{align*} similarly, \: 5 \times 5\times 5 &= 125 \\ or, \: 5^3 &= 125 \\ or, 5^{3 \times \frac {1}{3}} &= (125)^{\frac{1}{3}} \\ or, \: 5 &= (125)^{\frac{1}{3}} \\ or, \: 5 &= \sqrt[3]{125} \end{align*}
The cube root of 125 is 5.
In case of equation, root indicates the value of variable.
e.g. x + 2 = 0, x = -2,
So, -2 is the root of x
e.g. x2 - 4 = 0,
(x - 2)(x +2) = 0
Either, (x - 2) = 0 or, x = 2
or, (x + 2) = 0, x = -2
So, -2 and 2 are roots of x2 -4 =0
Surds are numerical expressions containing an irrational number. Surds may be quadratic, bi-quadratic, cubic etc.
For example: \( \sqrt{2}, \sqrt[3]{3}, \sqrt [4]{4}, \sqrt [5] {5}\)
The surds cannot be written in the form of \( \frac {p}{q} \) q≠ 0, so they are irrational numbers.
\(\boxed{\text {Note: Same process can be applied for division also}}\)
Surds are numerical expressions containing an irrational number. Surds may be quadratic, bi-quadratic, cubic etc.
For example: \( \sqrt2\), \(\sqrt[3]{2}\), \(\sqrt [4]{4}\), \(\sqrt [5] {5}\)
The surds cannot be written in the form of \( \frac {p}{q} \) q≠ 0, so they are irrational numbers.
.
Solution:
\begin{align*} &= 3\sqrt{2} + \sqrt [4] {2500} + \sqrt [4] {64} +6 \sqrt{8}\\&= 3\sqrt{2}+ \sqrt[4]{5 \times 5 \times 5 \times 5 \times 2 \times 2}+ \sqrt [4]{2 \times 2 \times 2 \times 2\times 2 \times 2}+ 6\sqrt{2 \times 2 \times 2}\\&= 3\sqrt{2}+5\sqrt[4]{4}+2\sqrt[4]{4}+12\sqrt{2}\\&=15\sqrt{2}+7\sqrt[4]{4} :_\text {Ans.} \end{align*}
\begin{align*} \sqrt [3] {16} + \sqrt [3] {54} - \sqrt [3] {250} &= \sqrt [3] {2^3 \times 2} + \sqrt [3] {3^3 \times 2} - \sqrt [3] {5^3 \times 2}\\ &= 2\sqrt [3]{2} + 3\sqrt [3]2 - 5\sqrt [3]2\\ &= 5\sqrt [3]2 - 5\sqrt [3]3\\ &= 0_{Ans}\end{align*}
\begin{align*} \frac {\sqrt {a + b} - \sqrt {a - b}}{\sqrt {a + b} + \sqrt {a - b}} &= \frac {\sqrt {a + b} - \sqrt {a - b}}{\sqrt {a + b} + \sqrt {a - b}} \times \frac {\sqrt {a + b} - \sqrt {a - b}}{\sqrt {a + b} - \sqrt {a - b}}\\ &= \frac {(\sqrt {a + b} - \sqrt {a - b})^2}{(\sqrt {a + b})^2 - (\sqrt {a - b})^2}\\ &= \frac {a + b + a - b - 2 \sqrt {a + b}\sqrt {a - b}}{a + b - a + b}\\ &= \frac {2a - 2\sqrt {a^2 - b^2}}{2b}\\ &= \frac {2(a - \sqrt {a^2 - b^2})}{2b}\\ &= \frac {a - \sqrt {a^2 - b^2}}b_{Ans}\\ \end{align*}
\(\frac {\sqrt y + \sqrt 5}{\sqrt y - \sqrt 5}\) = 3
or, \(\sqrt y\) + \(\sqrt 5\) = 3\(\sqrt y\) - 3\(\sqrt 5\)
or, 3\(\sqrt y\) - \(\sqrt y\) = \(\sqrt 5\) + 3\(\sqrt 5\)
or, 2\(\sqrt y\) = 4\(\sqrt 5\)
or, \(\sqrt y\) = \(\frac {4\sqrt 5}{2}\)
or, \(\sqrt y\) = 2\(\sqrt 5\)
Squaring on both sides,
(\(\sqrt y\))2= (2\(\sqrt 5\))2
or, y = 4× 5
∴ y = 20Ans
\(\frac {2y + 3}{\sqrt y - 1}\) = \(\frac 13\) (\(\sqrt y\) + 1)
or, 6y + 9 = (\(\sqrt y\) + 1) (\(\sqrt y\) - 1)
or, 6y + 9 = (\(\sqrt y\))2 - (1)2
or, 6y + 9 = y - 1
or, 6y - y = -1 -9
or, 5y = - 10
or, y = \(\frac {-10}5\)
∴ y = -2Ans
\(\sqrt x\) + 1 = 5 - \(\frac {\sqrt x -1}2\)
or, \(\sqrt x\) + 1 - 5 = \(\frac {- (\sqrt x - 1)}{2}\)
or, \(\sqrt x\) - 4 = \(\frac {-\sqrt x + 1}{2}\)
or, 2\(\sqrt x\) - 8 + \(\sqrt x\) = 1
or, 3\(\sqrt x\) = 1 + 8
or, 3\(\sqrt x\) = 9
or, \(\sqrt x\) = \(\frac 93\)
or, \(\sqrt x\) = 3
Squaring on both sides,
(\(\sqrt x\))2 = 32
∴ x = 9 Ans
\(\sqrt x\) + \(\sqrt {x - 20}\) = 10
or, \(\sqrt {x - 20}\) = 10 - \(\sqrt x\)
Squaring on both sides,
(\(\sqrt {x - 20}\))2 = (10 - \(\sqrt x\))2
or, x - 20 = 100 - 20\(\sqrt x\) + x
or, 20\(\sqrt x\) = 100 + 20
or, \(\sqrt x\) = \(\frac {120}{20}\)
or, \(\sqrt x\) = 6
Squaring on both sides,
(\(\sqrt x\))2 = 62
∴ x = 36Ans
\(\sqrt {4 (x + 1)}\) = \(\sqrt {4x}\) + 1
Squaring on both sides,
(\(\sqrt {4 (x + 1)}\))2 = (\(\sqrt {4x}\) + 1)2
or, 4(x + 1) = 4x + 2\(\sqrt {4x}\) + 1
or, 4x + 4 - 4x - 1 = 2\(\sqrt {4x}\)
or, \(\frac 32\) = \(\sqrt {4x}\)
Again,
Squaring on both sides,
(\(\sqrt {4x}\))2 = (\(\frac 32\))2
or, 4x = \(\frac 94\)
or, x = \(\frac 9{4 \times 4}\)
∴ x = \(\frac 9{16}_{Ans}\)
√27 + √75 - 8√3
six
five
eight
zero
√32 + √8 - √72
two
five
zero
one
√12 - √75 + √48
√3
√8
√7
√2
√50 + √18 - 8√2
five
nine
zero
three
√125 - √45 + √5
3√9
3√2
3√6
3√5
(sqrt {3x+13}=5)
2
8
4
5
(sqrt {2x+1}-3=0)
9
4
7
2
(sqrt {x+3}-1=2)
9
10
6
8
(sqrt {x+2}=3)
7
9
5
2
(sqrt {2a-1}=3)
5
2
12
9
4(sqrt {a-3}=sqrt{5a+7})
(frac{x-1}{sqrt{x}-1}=2+frac{sqrt x-5}{3})
(frac{x-6}{sqrt{x}-1}=4-frac{sqrt x-2}{3})
(frac{3x-1}{sqrt{3x+6}}=1+frac{sqrt 3x-1}{2})
(sqrt{x+3} ={sqrt{2x+4}}-1)
ASK ANY QUESTION ON Roots and Surds
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