CRE 3 - Square root of Surds | Algebra - Surds & Indices

Answer: Option C

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Explanation :

Square root of 7 + 4√3 will be of the form √x + √y.

∴ (√x + √y)² = 7 + 4√3

⇒ x + y + 2√xy = 7 + 4√3

Now, rational part of RHS = rational part on LHS
⇒ x + y = 7   …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ 2√xy = 4√3
⇒ xy = 12   …(2)

Solving (1) and (2), we get
x = 3 and y = 4 or x = 4 and y = 3.

In both cases square root of 7 + 4√3 = 2 + √3

Hence, option (c).

Answer: Option B

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Explanation :

Square root of 3 - 2√2 will be of the form √x - √y.

∴ (√x - √y)² = 3 - 2√2

⇒ x + y - 2√xy = 3 - 2√2

Now, rational part of RHS = rational part on LHS
⇒ x + y = 3   …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ 2√xy = 2√2
⇒ xy = 2   …(2)

Solving (1) and (2), we get
x = 2 and y = 1 or x = 1 and y = 2.

The square root of 3 - 2√2 will be either √2 - 1 or 1 - √2.

We will reject 1 - √2 as it is negative.

Hence, option (b).

Answer: Option C

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Explanation :

Square root of 14 + 6√5 will be of the form √x + √y.

∴ (√x + √y)² = 14 + 6√5

⇒ x + y + 2√xy = 14 + 6√5

Now, rational part of RHS = rational part on LHS
⇒ x + y = 14   …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ 2√xy = 6√5
⇒ xy = 45   …(2)

Solving (1) and (2), we get
x = 9 and y = 5 or x = 5 and y = 9.

In both cases square root of 14 + 6√5 = 3 + √5

Hence, option (c).

Answer: Option A

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Explanation :

√128 - √56 = √2 × (√64 - √28) = √2 × (8 - 2√7)

Now, square root of √2 is $\sqrt[4]{2}$

Let us calculate square root of (8 - 2√7)

Square root of 8 - 2√7 will be of the form √x - √y.

∴ (√x - √y)2 = 8 - 2√7

⇒ x + y - 2√xy = 8 - 2√7

Now, rational part of RHS = rational part on LHS
⇒ x + y = 8   …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ - 2√xy = - 2√7
⇒ xy = 7   …(2)

Solving (1) and (2), we get
x = 1 and y = 7 or x = 7 and y = 1.

The square root of 8 - 2√7 will be either √7 - 1 or 1 - √7

We will reject 1 - √7 as it is negative

Square root of √128 - √56 = $\sqrt[4]{2}\times \left(\sqrt{7}-1\right)$

Hence, option (a).

Answer: Option C

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Explanation :

Square root of 5 + √21 will be of the form √x + √y.

∴ (√x + √y)² = 5 + √21

⇒ x + y + 2√xy = 5 + √21

Now, rational part of RHS = rational part on LHS
⇒ x + y = 5   …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ 2√xy = √21
⇒ xy = 21/4   …(2)

⇒ x(5 - x) = 21/4 [From (1)]
⇒ 4x² - 20x + 21 = 0
⇒ x = 3/2 or 7/2

∴ x = 7/2 and y = 3/2 or x = 3/2 and y = 7/2.

In both cases square root of 5 + √21 = $\sqrt{\frac{7}{2}}+\sqrt{\frac{3}{2}}$

Hence, option (c).