## 17th SSC CHSL Solved Question Set, 3rd on Surds and indices

This is the 17th solved question set of 10 practice problem exercise for SSC CHSL exam and the 3rd on Surds and indices. You will find these 10 questions to be challenging.

It contains,

**Question set on Surds and Indices**for SSC CHSL to be answered in 15 minutes (10 chosen questions),**Answers**to the questions, and- Detailed
**conceptual solutions**to the questions. Main focus is on how quickly you get the answer.

For best results, take the test first, and then go through the solutions.

**Remember:** To be able to always answer any question on surds and indices quickly and confidently, you must solve many problems in a systematic manner using the conceptual analytical approach highlighted in these solutions.

### 17th Question set - 10 problems for SSC CHSL exam: 3rd on Surds and Indices - answering time 15 mins

**Q1. **The simplified value of $\displaystyle\frac{3\sqrt{2}}{\sqrt{3}+\sqrt{6}}-\displaystyle\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}}+\displaystyle\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}$ is,

- $\sqrt{2}$
- $0$
- $\displaystyle\frac{1}{\sqrt{2}}$
- $\sqrt{3}-\sqrt{2}$

**Q2.** What would be the remainder when $10^6-12$ is divided by 9?

- $7$
- $3$
- $4$
- $5$

**Q3. **If $z=6-2\sqrt{3}$, find the value of $\left(\sqrt{z}-\displaystyle\frac{1}{\sqrt{z}}\right)^2$.

- $\displaystyle\frac{12-46\sqrt{3}}{24}$
- $\displaystyle\frac{102-46\sqrt{3}}{4}$
- $\displaystyle\frac{102-46\sqrt{3}}{24}$
- $\displaystyle\frac{12-46\sqrt{3}}{4}$

**Q4. **Which of the following statement(s) is/are TRUE?

I. $\sqrt{676}+\sqrt{6.76}+\sqrt{0.0676}=27.76$

II. $\sqrt{339+\sqrt{36}+\sqrt{49}+\sqrt{81}}=19$

- Only II
- Only I
- Neither I nor II
- Both I and II

**Q5.** Let $\sqrt[3]{a}=\sqrt[3]{26}+\sqrt[3]{7}+\sqrt[3]{63}$. Then,

- $a \lt 729$ but $a \gt 216$
- $a \lt 216$
- $a \gt 729$
- $a=729$

**Q6.** Arrange the following in descending order,

$(\sqrt{23}-\sqrt{21})$, $(\sqrt{19}-\sqrt{17})$, $(\sqrt{21}-\sqrt{19})$.

- $(\sqrt{23}-\sqrt{21}) \gt (\sqrt{19}-\sqrt{17}) \gt (\sqrt{21}-\sqrt{19})$.
- $(\sqrt{23}-\sqrt{21}) \gt (\sqrt{21}-\sqrt{19}) \gt (\sqrt{19}-\sqrt{17})$.
- $(\sqrt{21}-\sqrt{19}) \gt (\sqrt{23}-\sqrt{21}) \gt (\sqrt{19}-\sqrt{17})$.
- $(\sqrt{19}-\sqrt{17}) \gt (\sqrt{21}-\sqrt{19}) \gt (\sqrt{23}-\sqrt{21})$.

**Q7.** If $\sqrt{21}=4.58$, what is the simplified value of $\left(8\sqrt{\displaystyle\frac{3}{7}}-3\sqrt{\displaystyle\frac{7}{3}}\right)$?

- $1$
- $0.474$
- $0.752$
- $0.655$

**Q8.** If $9^x=\sqrt[11]{243}$, then what is the value of $x$?

- $\displaystyle\frac{5}{7}$
- $\displaystyle\frac{5}{33}$
- $\displaystyle\frac{5}{22}$
- $\displaystyle\frac{5}{11}$

**Q9.** Calculate the value of $\left(5^{\frac{1}{4}}-1\right)\left(5^{\frac{3}{4}}+5^{\frac{1}{2}}+5^{\frac{1}{4}}+1\right)$.

- $4$
- $5$
- $10$
- $25$

**Q10.** $\sqrt[3]{a}=\sqrt[3]{9}+\sqrt[3]{126}+\sqrt[3]{217}$, then which of the following is correct?

- $a \lt 2197$
- $a \lt 1728$
- $a \gt 2197$
- $a=2197$

### Answers to the questions

**Q1. Answer:** Option b: $0$.

**Q2. Answer:** Option a: $7$.

**Q3. Answer:** Option c: $\displaystyle\frac{102-46\sqrt{3}}{24}$.

**Q4. Answer:** Option a: Only II.

**Q5. Answer:** Option a: $a \lt 729$ but $a \gt 216$.

**Q6. Answer:** Option d : $(\sqrt{19}-\sqrt{17}) \gt (\sqrt{21}-\sqrt{19}) \gt (\sqrt{23}-\sqrt{21})$.

**Q7. Answer:** Option d: $0.655$.

**Q8. Answer:** Option c: $\displaystyle\frac{5}{22}$.

**Q9. Answer:** Option a: $4$.

**Q10. Answer:** Option c: $a \gt 2197$.

### 17th solution set - 10 problems for SSC CHSL exam: 3rd on Surds and Indices - answering time was 15 mins

**Q1. **The simplified value of $\displaystyle\frac{3\sqrt{2}}{\sqrt{3}+\sqrt{6}}-\displaystyle\frac{4\sqrt{3}}{\sqrt{6}+\sqrt{2}}+\displaystyle\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}$ is,

- $\sqrt{2}$
- $0$
- $\displaystyle\frac{1}{\sqrt{2}}$
- $\sqrt{3}-\sqrt{2}$

** Solution 1: Simplification of denominators by first surd term factoring and then by Surd rationalization**

Identify that you can **simplify the denominators** of the first and second denominator by factoring out $\sqrt{3}$ from the first and $\sqrt{2}$ from the second denominator. This is applying the powerful surd term factoring technique. Result is,

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

$=\displaystyle\frac{3\sqrt{2}}{\sqrt{3}(\sqrt{2}+1)}-\displaystyle\frac{4\sqrt{3}}{\sqrt{2}(\sqrt{3}+1)}+\displaystyle\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}$

$=\sqrt{6}\left[\displaystyle\frac{1}{(\sqrt{2}+1)}-\displaystyle\frac{2}{(\sqrt{3}+1)}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}\right]$, canceling out factors $\sqrt{3}$ in the first and $\sqrt{2}$ in the second term and factoring $\sqrt{6}$ out of all three terms.

Now is the time to rationalize the three denominators by multiplying and dividing the first, the second and the third terms respectively by, $(\sqrt{2}-1)$, $(\sqrt{3}-1)$ and $(\sqrt{3}-\sqrt{2})$,

$E=\sqrt{6}\left[(\sqrt{2}-1)-(\sqrt{3}-1)+(\sqrt{3}-\sqrt{2})\right]$

$=0$.

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

**Key concepts used: Denominator simplification -- Surd term factoring -- Surd rationalization** --

*.*

**Solving in mind****Q2.** What would be the remainder when $10^6-12$ is divided by 9?

- $7$
- $3$
- $4$
- $5$

**Solution 2: Pattern identification and use of $10^6=99999+1$**

Mentally convert $10^6$ as $99999+1$ and subtract $12$ to get a simplified result,

$10^6-12=99999+1-12=99988$.

It is easy to see that the multiple of 9 nearest to 99988 and lower than it is 99981.

So if you divide 99988 by 9 remainder would be 7.

**Answer:** Option a: $7$.

**Key concepts used:** **Factors and multiples -- Key pattern identification -- Division and remainder concepts - Solving in mind****.**

**Q3. **If $z=6-2\sqrt{3}$, find the value of $\left(\sqrt{z}-\displaystyle\frac{1}{\sqrt{z}}\right)^2$.

- $\displaystyle\frac{12-46\sqrt{3}}{24}$
- $\displaystyle\frac{102-46\sqrt{3}}{4}$
- $\displaystyle\frac{102-46\sqrt{3}}{24}$
- $\displaystyle\frac{12-46\sqrt{3}}{4}$

**Solution 3: Key pattern identification to eliminate $\sqrt{z}$ from target and then Surd rationalization**

Your first thought may be—how would I simplify the given square root of surd $\sqrt{z}$! It doesn't look to be a promising approach at all. Isn't it?

Without wasting time on this track, when you look at the target expression, you could easily see that you can eliminate $\sqrt{z}$ just by simplifying the expression,

$E=\left(\sqrt{z}-\displaystyle\frac{1}{\sqrt{z}}\right)^2$

$=\displaystyle\frac{(z-1)^2}{z}$

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

It is dead easy to reach this point mentally. But even though full solution in mind is possible, it is better to scribble down at least the next step for ensuring correctness.

In the next step, rationalize the surd denominator by multiplying and dividing with $(6+2\sqrt{3})$,

$E=\displaystyle\frac{(37-20\sqrt{3})(6+2\sqrt{3})}{36-12}$

The denominator is 24 and the numeric part of the numerator can be mentally calculated to be, $222-120=102$.

So **you know the answer with certainty** as, $\displaystyle\frac{102-46\sqrt{3}}{24}$, because the **numerator surd terms** in the four choice options **are all $46\sqrt{3}$.**

Still for clarity remaining steps are given below,

$E=\displaystyle\frac{(37-20\sqrt{3})(6+2\sqrt{3})}{36-12}$

$=\displaystyle\frac{(222-120-120\sqrt{3}+74\sqrt{3})}{24}$

$=\displaystyle\frac{(102-46\sqrt{3})}{24}$.

**Answer:** Option c: $\displaystyle\frac{102-46\sqrt{3}}{24}$.

**Key concepts used: Pattern identification to eliminate square root of surd - Surd rationalization -- Identifying the second pattern of equal surd term in all choices to calculate the numeric part of the numerator quickly in mind for answer.**

**Q4. **Which of the following statement(s) is/are TRUE?

I. $\sqrt{676}+\sqrt{6.76}+\sqrt{0.0676}=27.76$

II. $\sqrt{339+\sqrt{36}+\sqrt{49}+\sqrt{81}}=19$

- Only II
- Only I
- Neither I nor II
- Both I and II

**Solution 4: Converting to equal bases of number under square root and knowledge of squares of two digit numbers**

For the first option, mentally convert the second term to,

$\sqrt{6.76}=0.1\times{\sqrt{676}}=2.6$, **as square root of 676 equals 26.**

Similarly convert the third term to,

$\sqrt{0676}=0.01\times{\sqrt{676}}=0.26$.

So, the LHS of the first option evaluates to,

$\sqrt{676}+\sqrt{6.76}+\sqrt{0.0676}=26+2.6+0.26=28.86 \neq 27.76$, the Statement I is FALSE.

The LHS of the second Statement II evaluates to,

$\sqrt{339+\sqrt{36}+\sqrt{49}+\sqrt{81}}$

$=\sqrt{339+6+7+9}=\sqrt{361}=19=RHS$.

Only the second statement is TRUE.

**Answer:** Option a: Only II.

**Key concepts used: Square root of two digit numbers -- Converting base to equal values of number under square root -- Base equalization technique -- Solving in mind.**

**Q5.** Let $\sqrt[3]{a}=\sqrt[3]{26}+\sqrt[3]{7}+\sqrt[3]{63}$. Then,

- $a \lt 729$ but $a \gt 216$
- $a \lt 216$
- $a \gt 729$
- $a=729$

**Solution 5: Problem analysis, Key pattern identification and Basic Inequality Concepts**

To get the answer you need to raise $\sqrt[3]{a}$ to the power of 3. But with three RHS terms in the equation it is infeasible. So what do you do?

Then you discover the interesting **pattern that each of the RHS terms has numbers under cube root less than 1 of a perfect cube root**,

$\sqrt[3]{26}=\sqrt[3]{27-1}$

$\sqrt[3]{7}=\sqrt[3]{8-1}$

$\sqrt[3]{63}=\sqrt[3]{64-1}$.

Now it is easy to form first an inequality of RHS against an integer and so with $\sqrt[3]{a}$,

$RHS=\sqrt[3]{27-1}+\sqrt[3]{8-1}+\sqrt[3]{64-1} \lt (\sqrt[3]{27}+\sqrt[3]{8}+\sqrt[3]{64})$

Or, $RHS \lt (3+2+4)$,

Or, $\sqrt[3]{a} \lt 9$.

Now take cube of both sides of the inequality.

$a \lt 729$.

But from the RHS values it can also be deduced easily that,

$\sqrt[3]{a} \gt 6$,

Or, $a \gt 216$.

**Note:** $\sqrt[3]{a} \gt 6$ because,

$\sqrt[3]{a}=\sqrt[3]{26}+\sqrt[3]{7}+\sqrt[3]{63}$

$=2.x+1.y+3.z=6.p \gt 6$, where $p=x+y+z \gt 0$.

**Answer:** Option a: $a \lt 729$ but $a \gt 216$.

*Key concepts used:* **Key pattern identification -- Converting RHS numbers under cube root in terms of perfect cube roots -- Basic Inequality concepts -- Solving in mind****.**

Yes, if you can identify the all important key pattern you can also solve the problem easily and quickly in mind.

**Q6.** Arrange the following in descending order,

$(\sqrt{23}-\sqrt{21})$, $(\sqrt{19}-\sqrt{17})$, $(\sqrt{21}-\sqrt{19})$.

- $(\sqrt{23}-\sqrt{21}) \gt (\sqrt{19}-\sqrt{17}) \gt (\sqrt{21}-\sqrt{19})$.
- $(\sqrt{23}-\sqrt{21}) \gt (\sqrt{21}-\sqrt{19}) \gt (\sqrt{19}-\sqrt{17})$.
- $(\sqrt{21}-\sqrt{19}) \gt (\sqrt{23}-\sqrt{21}) \gt (\sqrt{19}-\sqrt{17})$.
- $(\sqrt{19}-\sqrt{17}) \gt (\sqrt{21}-\sqrt{19}) \gt (\sqrt{23}-\sqrt{21})$.

**Solution 6: Problem analysis and two term surd expression comparison by** Equal difference surd comparison technique

The Equal difference surd comparison technique states,

In a number of two term subtractive (example: $\sqrt{7}-\sqrt{5}$) surd expressions compared, if the difference between the numbers ($7-5=2$ in the example) under the square root for each expression is same, the expression with the first term that has

highest such number under square root will be the lowest.

For example,

$(\sqrt{23}-\sqrt{21}) \lt (\sqrt{19}-\sqrt{17})$, as $23-21=2$, $19-17=2$, and $23 \gt 19$.

#### Proof of Equal difference surd comparison technique

$(\sqrt{23}+\sqrt{21}) \gt (\sqrt{19}+\sqrt{17})$,

Or, $\displaystyle\frac{1}{(\sqrt{23}+\sqrt{21})} \lt \displaystyle\frac{1}{(\sqrt{19}+\sqrt{17})}$.

Rationalize the denominators and you get your result,

$(\sqrt{23}-\sqrt{21}) \lt (\sqrt{19}-\sqrt{17})$, $\frac{1}{2}$ on both sides cancels out.

Applying this powerful surd comparison technique it takes a very short time in mind to sequence the three given surd expressions in descending order as,

$(\sqrt{19}-\sqrt{17}) \gt (\sqrt{21}-\sqrt{19}) \gt (\sqrt{23}-\sqrt{21})$.

**Answer:** Option d : $(\sqrt{19}-\sqrt{17}) \gt (\sqrt{21}-\sqrt{19}) \gt (\sqrt{23}-\sqrt{21})$.

**Key concepts used:** *Equal difference surd comparison technique -- Inequality analysis -- Solving in mind.*

**Q7.** If $\sqrt{21}=4.58$, what is the simplified value of $\left(8\sqrt{\displaystyle\frac{3}{7}}-3\sqrt{\displaystyle\frac{7}{3}}\right)$?

- $1$
- $0.474$
- $0.752$
- $0.655$

**Solution 7: Problem analysis and Solving by Key pattern identification and Direct approach**

How to combine the two surd fractions leaving only $\sqrt{21}$ as surd in the denominator?

Thinking for a moment you decide to combine the two terms just like a normal fraction subtraction. Bur first you express the two terms in the form of two fractions to simplify the look,

$\left(8\sqrt{\displaystyle\frac{3}{7}}-3\sqrt{\displaystyle\frac{7}{3}}\right)$

$=\displaystyle\frac{8\sqrt{3}}{\sqrt{7}}-\displaystyle\frac{3\sqrt{7}}{\sqrt{3}}$

$=\displaystyle\frac{8\times{3}-3\times{7}}{\sqrt{21}}$

$=\displaystyle\frac{3}{4.58}$

$=\displaystyle\frac{300}{458}$

$=0.655$, dividing mentally 3000 by 458 to get first digit as 0.6 that identifies the choice uniquely.

**Answer:** Option d: $0.655$.

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

**Pattern identification of unique first digit 0.6 to cut short calculation time***--*Solving in mind.**Q8.** If $9^x=\sqrt[11]{243}$, then what is the value of $x$?

- $\displaystyle\frac{5}{7}$
- $\displaystyle\frac{5}{33}$
- $\displaystyle\frac{5}{22}$
- $\displaystyle\frac{5}{11}$

** Solution 8: Problem analysis and solving by equalizing both bases to 3**—Base equalization technique

Identify the pattern both the terms in LHS and RHS can be expressed in terms of powers of base 3. This is application of base equalization technique.

The given expression,

$9^x=\sqrt[11]{243}$

Or, $3^{2x}=3^{\frac{5}{11}}$.

As the bases are equal to 3, powers of both the terms must also be equal,

$2x=\displaystyle\frac{5}{21}$,

Or, $x=\displaystyle\frac{5}{22}$.

**Answer:** Option c: $\displaystyle\frac{5}{22}$.

**Indices rule** used is,

If $a^p=a^q$, then $p=q$.

**Key concepts used:**** Base equalization technique -- Indices rule -- Solving in mind.**

**Q9.** Calculate the value of $\left(5^{\frac{1}{4}}-1\right)\left(5^{\frac{3}{4}}+5^{\frac{1}{2}}+5^{\frac{1}{4}}+1\right)$.

- $4$
- $5$
- $10$
- $25$

**Solution 9: Problem analysis and Solving by Key pattern identification and Abstraction by Substitution**

Identify the key pattern that,

$5^{\frac{3}{4}}=\left(5^{\frac{1}{4}}\right)^3$, and

$5^{\frac{1}{2}}=\left(5^{\frac{1}{4}}\right)^2$.

Decide that instead of dealing with the awkward fraction power terms it would be best to adopt abstraction by the substitution,

$x=5^{\frac{1}{4}}$.

This would immediately simplify the look of the target expression and make algebraic simplification of the target expression comfortable,

$\left(5^{\frac{1}{4}}-1\right)\left(5^{\frac{3}{4}}+5^{\frac{1}{2}}+5^{\frac{1}{4}}+1\right)$

$=(x-1)(x^3+x^2+x+1)$

$=x(x-1)(x^2+x+1)+(x-1)$, by second pattern identification of three-term factor of $(x^3-1)$,

$=x(x^3-1)+(x-1)$

$=x^4-x+x-1=x^4-1=5-1=4$.

**Answer:** Option a: $4$.

**Key concepts used:** * Key pattern identification in two stages -- Abstraction -- Substitution -- Two factor expansion of sum of cubes *--

**Solving in mind.**With the substitution and identification of possible use of expansion of $(x^3-1)$, solving in mind takes a few tens of seconds.

**Q10.** $\sqrt[3]{a}=\sqrt[3]{9}+\sqrt[3]{126}+\sqrt[3]{217}$, then which of the following is correct?

- $a \lt 2197$
- $a \lt 1728$
- $a \gt 2197$
- $a=2197$

**Solution 10: Problem analysis and Solution by Key pattern identification and Basic inequality concepts**

Raising the equation to its cube being infeasible for quick solution, you look for a hidden clue (that must be there you are sure).

When you examine the RHS of the given equation with specific intent, you discover quickly the key pattern that in the three terms of RHS,

$9=8+1=2^3+1$,

$126=125+1=5^3+1$, and,

$217=216+1=6^3+1$.

Using basic inequality concepts now it is easy for you to form the inequality relation of the RHS and hence LHS,

$\sqrt[3]{9}+\sqrt[3]{126}+\sqrt[3]{217}=\sqrt[3]{2^3+1}+\sqrt[3]{5^3+1}+\sqrt[3]{6^3+1}$

Or, $\sqrt[3]{a} \gt \sqrt[3]{2^3}+\sqrt[3]{5^3}+\sqrt[3]{6^3} \gt (2+5+6)$,

Or, $a \gt 13^3$, raising the inequality relation to its cube,

Or, $a \gt (169\times{13})$,

Or, $a \gt 2197$.

**Answer: **Option c: $a \gt 2197$.

**Key concepts used:**** Key pattern discovery -- Basic inequality concepts -- Solving in mind.**

This question consolidates the concepts and techniques used in Q5.

### End note

This solved question set under the category of quantitative aptitude tests should be useful for a wide variety of competitive tests such as, AMCAT, eLitmus, SSC Matric level, PSC, RRB, TET, SBI Clerk, IBPS Clerk, LIC, AAO, RBI, AFCAT, UPSC CDS, Bank clerk and even SSC CGL Tier II.

### Guided help on Fractions, Surds and Indices in Suresolv

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