## 7th Solution set on 10 Surds and Indices questions, 73rd for SSC CGL

Quick Solutions to 10 surds and indices questions of 7th practice set on the topic and 73rd for SSC CGL. Learn the techniques for solving such hard surds indices questions in details from **How to solve surds 2.**

This solution set should be *useful for all competitive exams with surds and indices.*

Take the test from * SSC CGL level Question Set 73, Surds and Indices 7* if you have not yet taken the test.

And to *clear all doubts on surds and indices problem solving*, go through the extensive coverage on the topic **from the guide** (with list of all tutorials, question and solution sets at Suresiolv),

**Guide to Fractions, Surds and Indices problem solving.**

### 7th Solution set on 10 Surds and Indices questions, 73rd for SSC CGL - answering time was 15 mins

#### Problem 1.

The value of $\displaystyle\frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}+\displaystyle\frac{\sqrt{12}}{\sqrt{3}-\sqrt{2}}$ is,

- $11$
- $-12$
- $12$
- $-11$

**Solution 1: Problem analysis and solving**

Take the direct approach and rationalize the two terms by multiplying and dividing the first by $(3\sqrt{2}-2\sqrt{3})$ and the second by $(\sqrt{3}+\sqrt{2})$. The denominators will become pure integers and the result,

$\displaystyle\frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}+\displaystyle\frac{\sqrt{12}}{\sqrt{3}-\sqrt{2}}$

$=\displaystyle\frac{(3\sqrt{2}-2\sqrt{3})^2}{18-12}+\sqrt{12}(\sqrt{3}+\sqrt{2})$

$=\displaystyle\frac{1}{6}\left(18+12-12\sqrt{6}\right)+6+2\sqrt{6}$

$=5-2\sqrt{6}+6+2\sqrt{6}$

$=11$.

**Answer:** Option a: $11$.

**Key concepts used:** * Surd rationalization -- Surd arithmetic* --

**.**

*Surd simplification*This solution needed writing a few steps but still could be finished comfortably within a minute because of the inherent simplicity of the steps as well as the assurance from choice values that the surd terms will cancel out.

#### Problem 2.

Value of $\displaystyle\frac{\sqrt{5}+\sqrt{3}}{\sqrt{80}+\sqrt{48}-\sqrt{45}-\sqrt{27}}$ is,

- $-2$
- $-1$
- $2$
- $1$

**Solution 2: Problem analysis and solving**

Identify the key pattern in the four surds in the denominator - each of the four surds has a factor that is a square, 16, 16, 9 and 9 respectively.

Apply surd term factoring by taking the squares out of the square roots,

$\displaystyle\frac{\sqrt{5}+\sqrt{3}}{\sqrt{80}+\sqrt{48}-\sqrt{45}-\sqrt{27}}$

$=\displaystyle\frac{\sqrt{5}+\sqrt{3}}{4\sqrt{5}+4\sqrt{3}-3\sqrt{5}-3\sqrt{3}}$

$=\displaystyle\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

$=1$.

The numerator and denominator cancel out.

**Answer.** Option d: $1$.

**Concepts.** * Key Pattern identification* --

*.*

**Surd term factoring**#### Problem 3.

Simplify $\displaystyle\frac{6}{2\sqrt{3}-\sqrt{6}}+\displaystyle\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}-\displaystyle\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}$ is,

- $1$
- $2$
- $0$
- $-1$

**Solution 3: Problem analysis and solving**

In such a problem with multiple surd fractions, we always look for two patterns or opportunities,

**Surd term factoring**, and/or,**Possible surd rationalization.**

With such **directed attention**, the first term could easily be simplified by first factoring $\sqrt{6}$ out of first term of the denominator,

$2\sqrt{3}=\sqrt{6}\times{\sqrt{2}}$,

And then factoring $\sqrt{6}$ out of the whole denominator and canceling it out with the numerator,

$\displaystyle\frac{6}{2\sqrt{3}-\sqrt{6}}$

$=\displaystyle\frac{\sqrt{6}}{\sqrt{2}-1}$

$=\sqrt{6}(\sqrt{2}+1)$, by rationalization.

This is both **Surd term factoring and rationalization of surds in action.**

Next the second term is rationalized by multiplying and dividing it with $\sqrt{3}-\sqrt{2}$. Result is,

$\displaystyle\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}$

$=\sqrt{6}(\sqrt{3}-\sqrt{2})$.

Surd term factoring could again be applied on the third term by factoring out $\sqrt{2}$ from the numerator as well as both terms of the denominator thus canceling it out. Resiult is,

$\displaystyle\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}$

$=\displaystyle\frac{2\sqrt{6}}{\sqrt{3}-1}$

$=\sqrt{6}(\sqrt{3}+1)$, by surds rationalization.

In all three terms, the factor $\sqrt{6}$ is kept as it is without further multiplying it with the terms inside the brackets because it could be seen that this factor will be common to all three terms.

Now add up the three simplified results and get the value of the target expression as,

$E=\sqrt{6}(\sqrt{2}+1+\sqrt{3}-\sqrt{2}-\sqrt{3}-1)=0$.

Simplification in a few steps depended upon the suitable

surd term factoringfor canceling out surd factors between the numerators and denominators as well as the strategy of first simplifying the denominator and then only rationalize the surd denominators.

**Answer:** Option c: $0$.

**Key concepts used: ****Surd term factoring -- ****Denominator simplification first and then Surd rationalization.**

#### Problem** 4.**

Simplify $\displaystyle\frac{4\sqrt{18}}{\sqrt{12}}-\displaystyle\frac{8\sqrt{75}}{\sqrt{32}}+\displaystyle\frac{9\sqrt{2}}{\sqrt{3}}$.

- $2$
- $0$
- $-1$
- $1$

#### Solution** 4: Problem analysis and solving**

Applying surd term factoring on all the terms,

$\displaystyle\frac{4\sqrt{18}}{\sqrt{12}}-\displaystyle\frac{8\sqrt{75}}{\sqrt{32}}+\displaystyle\frac{9\sqrt{2}}{\sqrt{3}}$

$=\displaystyle\frac{12\sqrt{2}}{2\sqrt{3}}-\displaystyle\frac{40\sqrt{3}}{4\sqrt{2}}+\displaystyle\frac{9\sqrt{2}}{\sqrt{3}}$

$=2\sqrt{6}-5\sqrt{6}+3\sqrt{6}$, this time we have split the integer factors into square of surd terms for cancelling the surd denominators; this is reverse surd term factoring

$=0$.

**Answer:** Option b: $0$.

**Key concepts used:** ** Fraction arithmetic **--

**.**

*Surd term factoring -- Reverse surd term factoring -- denominator elimination*#### Reverse surd term factoring

An example of reverse surd term factoring,

$9\sqrt{2}=3(\sqrt{3})^2\sqrt{2}=3\sqrt{3}\sqrt{6}$.

This tactic is specially employed to break up a normal surd term with integer coefficient into square of surds to bring out a surd factor present in the denominator for eliminating the denominator.

#### Problem** 5.**

Value of

$\displaystyle\frac{1}{1+\sqrt{2}}+\displaystyle\frac{1}{\sqrt{2}+\sqrt{3}}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{4}}$

$+\displaystyle\frac{1}{\sqrt{4}+\sqrt{5}}+\displaystyle\frac{1}{\sqrt{5}+\sqrt{6}}+\displaystyle\frac{1}{\sqrt{6}+\sqrt{7}}$

$+\displaystyle\frac{1}{\sqrt{7}+\sqrt{8}}+\displaystyle\frac{1}{\sqrt{8}+\sqrt{9}}$ is,

- $0$
- $-2$
- $2$
- $1$

#### Solution 5: Problem analysis and solving execution

Two key patterns identified are,

- Difference between the surd term values under square roots in each denominator is 1, and
- Adjacent pairs of denominator surd terms have a common surd term.

Rationalizing the terms results in canceling out all the intermediate surd terms leaving one from the first pair and one from the last pair.

Result thus is,

$E=3-1=2$.

**Answer:** Option c: $2$.

**Key concepts used:** **Pattern identification -- Surd rationalization.**

Let us show the deductive steps.

Rationalizing all the denominators and taking care of placing the larger term in each pair first in the subtractive numerator expressions, the target expression is transformed to,

$E=(\sqrt{2}-1)+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})$

$+(\sqrt{5}-\sqrt{4})+(\sqrt{6}-\sqrt{5})+(\sqrt{7}-\sqrt{6})$

$+(\sqrt{8}-\sqrt{7})+(\sqrt{9}-\sqrt{8})$

$=\sqrt{9}-1$

$=2$.

#### Problem 6.

If $a=\displaystyle\frac{1}{3+2\sqrt{2}}$ and $b=\displaystyle\frac{1}{3-2\sqrt{2}}$, the value of $a^2b +ab^2$ is,

- $-5$
- $6$
- $-6$
- $5$

#### Solution** 6: Problem analysis and solving execution**

First we identify the two patterns,

$ab=\displaystyle\frac{1}{3+2\sqrt{2}}\times{\displaystyle\frac{1}{3-2\sqrt{2}}}$

$=\displaystyle\frac{1}{9-8}$

$=1$, and,

$a+b=\displaystyle\frac{1}{3+2\sqrt{2}}+\displaystyle\frac{1}{3-2\sqrt{2}}$

$=6$, the denominator is simplified to 1, and the surd terms cancel out in the numerator.

Turning our attention to the target expression the third pattern is identified as,

$a^2b+ab^2=ab(a+b)$.

So the answer turns out to be 6.

**Answer:** Option b: $6$.

**Key concepts used:** * Pattern identification* --

*.*

**Surd simplification**Done easily in mind. Quick solution depended on the similar pattern identification and use.

#### Problem** 7.**

If $x=\displaystyle\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ and $y=\displaystyle\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$, the value of $x^3+y^3$ is,

- 807
- 907
- 870
- 970

#### Solution 7: Problem analysis and solving execution

The patterns identified in the given expressions are simple value of $xy$ and $x+y$,

$xy=1$, and

$x+y=(\sqrt{3}-\sqrt{2})^2+(\sqrt{3}+\sqrt{2})^2$

$=10$.

Knowing that the sum of cubes expands to,

$x^3+y^3=(x+y)(x^2-xy+y^2)$,

we express it in terms of $xy$ and $x+y$,

$x^3+y^3=(x+y)\left[(x+y)^2-3xy\right]$

$=10\times{(100-3)}$

$=970$.

**Answer:** Option d: 970.

**Key concepts used:** ** Pattern identification **--

*input driven target transformation -- Surd algebra.*#### Problem** 8.**

If $x=\sqrt{\displaystyle\frac{5+2\sqrt{6}}{5-2\sqrt{6}}}$ find the value of $x^2(x-10)^2$,

- $-1$
- $-2$
- $1$
- $2$

#### Solution** 8: Problem analysis and strategy decision**

First key pattern we identify,

$5 + 2\sqrt{6}=(\sqrt{3}+\sqrt{2})^2$, and

$5-2\sqrt{6}=(\sqrt{3}-\sqrt{2})^2$.

We can then easily eliminate the covering square root of the surd expression.

But as target expression has $x^2$ as the first product term, we desist from this temptation and remove the covering square root just by simple squaring,

$x^2=\displaystyle\frac{5+2\sqrt{6}}{5-2\sqrt{6}}$.

The reason for this simple approach is visible now—we need to rationalize and as soon as we do that, the RHS is transformed to a square of surd sum,

$x^2=(5+2\sqrt{6})^2$.

If we had removed the covering square root earlier, the power of the surd expression would have been 4 now.

#### Solution 8: Problem solving second stage

Turning our attention to the second product term, $(x-10)^2$ now we feel the need of using the value of $x$ without covering square root,

$x-10=\displaystyle\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}-10$

$=(\sqrt{3}+\sqrt{2})^2-10$

$=-5+2\sqrt{6}$.

So value of target expression value wil be,

$x^2(x-10)^2$

$=\left[(5+2\sqrt{6})(5-2\sqrt{6})\right]^2$, the negative sign is nullified by the squaring

$=1$.

**Answer:** Option c: 1.

**Key concepts used:** * Problem analysis* --

*--*

**Problem solving strategy decision***.*

**Surd rationalization -- Double square root surd expressions -- Analytical approach example**#### Problem** 9.**

Arrange $3^{34}$, $2^{51}$, and $7^{17}$ in ascending order,

- $3^{34}$ > $2^{51}$ > $7^{17}$
- $2^{51}$ > $3^{34}$ > $7^{17}$
- $3^{34}$ > $7^{17}$ > $2^{51}$
- $7^{17}$ > $2^{51}$ > $3^{34}$

#### Problem** analysis and solving execution by base equalization**

Identifying the pattern that the powers are in factors multiples relation we decide to equalize the powers to the lowest power value of 17,

$2^{51}=8^{17}$

$3^{34}=9^{17}$.

The third one is, $7^{17}$.

Power 17 being the same, with highest value 9, $3^{34}$ is the highest of the three, and second highest is $2^{51}$.

The three arranged in ascending order is,

$3^{34}$ > $2^{51}$ > $7^{17}$.

**Answer.** Option a: $3^{34}$ > $2^{51}$ > $7^{17}$.

**Key concepts used:** **Pattern identification -- Indices -- Base equalization technique, in this case power is equalized.**

Could be carried out in mind.

#### Problem** 10.**

$4^{61}+4^{62}+4^{63}+4^{64}$ is divisible by,

- $3$
- $11$
- $13$
- $17$

#### Solution 10: Problem analysis and solving

Taking out $4^{61}$ as the common factor the given expression turns to,

$4^{61}+4^{62}+4^{63}+4^{64}$

$=4^{61}(1+4+4^2+4^3)$

$=4^{61}(85)$, divisible by 17.

**Answer:** Option d: 17.

**Key concepts used:** ** Factoring out large power term**.

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