## Surds and indices questions for SSC CHSL 1st set, Answers and Quick Solutions

1st set of surds and indices questions for SSC CHSL with answers and quick solutions by surds techniques. These are previous year SSC CHSL questions.

The set contains,

**Surds and Indices questions**for SSC CHSL to be answered in 15 minutes (10 chosen questions)**Answers**to the questions, and**Quick conceptual solutions**to the questions.

Take the timed test, verify answers and learn how to solve the questions quickly from solutions.

### 1st set of 10 Surds and indices questions for SSC CHSL - answering time 15 mins

**Q1. **The total number of prime factors in $4^{10}\times{7^3}\times{16^2}\times{11}\times{10^2}$ is,

- 36
- 34.
- 37
- 35

**Q2.** If $x^{\frac{1}{4}}+x^{-\frac{1}{4}}=2$ then what is the value of $x^{\frac{1}{81}}+x^{-\frac{1}{81}}$?

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

**Q3. **If $x$, $y$ are rational numbers and $\displaystyle\frac{5+\sqrt{11}}{3-2\sqrt{11}}=x+y\sqrt{11}$, the values of $x$ and $y$ are,

- $x=\displaystyle\frac{4}{13}$, $y=\displaystyle\frac{11}{17}$
- $x=-\displaystyle\frac{14}{17}$, $y=-\displaystyle\frac{13}{26}$
- $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$
- $x=-\displaystyle\frac{27}{25}$, $y=-\displaystyle\frac{11}{37}$

**Q4. **The greatest among the numbers, $0.16$, $\sqrt{0.16}$, $(0.16)^2$, $0.04$ is,

- $0.16$
- $\sqrt{0.16}$
- $(0.16)^2$
- $0.04$

**Q5.** Out of the numbers 0.3, 0.03, 0.9 and 0.09, the number that is nearest to the value of $\sqrt{0.9}$ is,

- $0.09$
- $0.3$
- $0.03$
- $0.9$

**Q6.** Which of the following statement(s) is/are TRUE?

- $\sqrt{144}\times{\sqrt{36}} \lt \sqrt[3]{125}\times{\sqrt{121}}$
- $\sqrt{324}+\sqrt{49} \lt \sqrt[3]{216}\times{\sqrt{9}}$

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

**Q7.** If $\displaystyle\frac{(x-\sqrt{24})(\sqrt{75}+\sqrt{50})}{\sqrt{75}-\sqrt{50}}=1$, then the value of $x$ is,

- $3\sqrt{5}$
- $5$
- $\sqrt{5}$
- $2\sqrt{5}$

**Q8.** The value of $\displaystyle\frac{1}{\sqrt{2}+1}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}+\displaystyle\frac{1}{\sqrt{4}+\sqrt{3}}+.....+\displaystyle\frac{1}{\sqrt{100}+\sqrt{99}}$ is,

- $1$
- $\sqrt{99}$
- $9$
- $\sqrt{99}-1$

**Q9.** If $1^3+2^3+3^3....+10^3=3025$, then the value of $2^3+4^3+....+20^3$ is,

- 7590
- 5060
- 12100
- 24200

**Q10.** $\left(\displaystyle\frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}}+\displaystyle\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}\right)$ simplifies to,

- $\sqrt{5}-\sqrt{6}$
- $\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}-3\sqrt{6}$

### Answers to the1st Set of Surds and indices questions for SSC CHSL

**Q1. Answer:** Option a: 36.

**Q2. Answer:** Option c: $2$.

**Q3. Answer:** Option c: $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$

**Q4. Answer:** Option b: $\sqrt{0.16}$.

**Q5. Answer:** Option d: $0.9$.

**Q6. Answer:** Option d : Neither I nor II.

**Q7. Answer:** Option b: 5.

**Q8. Answer:** Option c: 9.

**Q9. Answer:** Option d: 24200.

**Q10. Answer:** Option a: $\sqrt{5}-\sqrt{6}$.

### Solutions to the 1st Set of Surds and Indices questions for SSC CHSL - answering time was 15 mins

**Q1. **The total number of prime factors in $4^{10}\times{7^3}\times{16^2}\times{11}\times{10^2}$ is,

- 36
- 34.
- 37
- 35

** Solution 1: Problem analysis and solution by prime factorization of each term of the product**

**Method:** The prime numbers in the product to be identified. Each of the primes to be expressed as raised to suitable power. Sum of the powers will be the desired number of prime factors.

Let's first take care of prime base factor 2.

$4^{10}=2^{20}$ contributes to 20 number of 2s.

$16^2=2^8$ contributes to 8 number of 2s, and

$10^2=2^2.5^2$ contributes to 2 number of 2s.

Total number of 2s is 30.

Add to this 2 number of 5s from $10^2$, 1 number of 11 and 3 number of 7s from $7^3$.

The total number of prime factors in the product is,

$30+2+1+3=36$.

**Answer:** Option a: 36.

**Key concepts used: Counting prime factors in a product -- Indices -- Prime factorization** --

*.*

**Solving in mind****Q2.** If $x^{\frac{1}{4}}+x^{-\frac{1}{4}}=2$ then what is the value of $x^{\frac{1}{81}}+x^{-\frac{1}{81}}$?

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

**Solution 2: Problem solving using principle of interaction of inverses**

You cannot derive the value of the target expresion with large inverse powers in $x$ from the given expression directly.

The only possibility is to get a simple value of $x$ from the given expression, which is expected to be 1, and use the value in the target expression.

The LHS of the given equation is a sum of inverses that can be expressed as a square of inverses in powers of $x$ as 2. This is easy because the middle term will be neutralized to an integer. Results as planned are,

$x^{\frac{1}{4}}+x^{-\frac{1}{4}}=2$,

Or, $(x^{\frac{1}{2}})^2-2.x^{\frac{1}{2}}.x^{-\frac{1}{2}}+(x^{-\frac{1}{2}})^2=0$,

Or, $(x^{\frac{1}{2}}-x^{-\frac{1}{2}})^2=0$,

So, $x^{\frac{1}{2}}=x^{-\frac{1}{2}}$,

Or, $x^{\frac{1}{4}}=1$,

So, $x=1$.

Answer is,

$x^{\frac{1}{81}}+x^{-\frac{1}{81}}=1+1=2$

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

**Key concepts used:** **Principle of interaction of sum of inverses -- Mathematical reasoning -- Indices-- Solving in mind****.**

**Q3. **If $x$, $y$ are rational numbers and $\displaystyle\frac{5+\sqrt{11}}{3-2\sqrt{11}}=x+y\sqrt{11}$, the values of $x$ and $y$ are,

- $x=\displaystyle\frac{4}{13}$, $y=\displaystyle\frac{11}{17}$
- $x=-\displaystyle\frac{14}{17}$, $y=-\displaystyle\frac{13}{26}$
- $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$
- $x=-\displaystyle\frac{27}{25}$, $y=-\displaystyle\frac{11}{37}$

**Solution 3: Problem analysis and solution by pattern identification, remainder concept and HCF concept**

**Method:**

You have to first simplify the LHS of the given equation by **eliminating the surd denominator** to get an expression of the form, $a+b\sqrt{11}$.

Now compare the like terms (irrational surd and rational) on two sides of the equation to get the values of $x$ and $y$.

**Execute:**

To eliminate the surd expression in the denominator, rationalize it by multiplying both numerator and denominator by $(3+2\sqrt{11})$. Result is,

$\displaystyle\frac{5+\sqrt{11}}{3-2\sqrt{11}}\times{\displaystyle\frac{3+2\sqrt{11}}{3+2\sqrt{11}}}=x+y\sqrt{11}$,

Or, $-\displaystyle\frac{15+22+13\sqrt{11}}{35}=x+y\sqrt{11}$,

Or, $-\displaystyle\frac{37}{35}-\displaystyle\frac{13}{35}.\sqrt{11}=x+y\sqrt{11}$.

Equate rational term $x$ with $-\displaystyle\frac{37}{35}$ and **rational coefficients of irrational surd** $\sqrt{11}$ on both sides of the equation (becauseuse irrational and rational numbers can't be added together),

$x=-\displaystyle\frac{37}{35}$, and

$y=-\displaystyle\frac{13}{35}$.

**Answer:** Option c: $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$.

**Key concepts used: Rationalization of surd denominator -- Comparing rational coefficients of surd terms and equating rational terms on both sides of a surd equation.**

**Q4. **The greatest among the numbers, $0.16$, $\sqrt{0.16}$, $(0.16)^2$, $0.04$ is,

- $0.16$
- $\sqrt{0.16}$
- $(0.16)^2$
- $0.04$

**Solution 4: Problem solving using concept of powers of a decimal number**

The **basic results of raising a decimal number to a power** are,

If a decimal that is less than 1, is raised to a positive integer power, the value of the decimal with power gets smaller than the original decimal number. For example, $(0.4)^2=0.16 \lt 0.4$.

Similarly, if a decimal that is less than 1, is raised to a positive power less than 1, the value of the decimal in power increases. For example, $(0.25)^{\frac{1}{2}}=0.5 \gt 0.25$.

Applying this pair of concepts, identify $\sqrt{0.16}$ as the largest among the first three given numbers, $0.16$, $\sqrt{0.16}$ and $(0.16)^2$.

Now take the square root of $0.16$ to compare it with the fourth number $0.04$,

$\sqrt{0.16}=0.4 \gt 0.04$.

So, $\sqrt{0.16}$ is the largest among the four given numbers.

**Answer:** Option b: $\sqrt{0.16}$.

**Key concepts used: Principle of powers of decimal numbers -- Square root of a decimal number -- Solving in mind.**

**Q5.** Out of the numbers 0.3, 0.03, 0.9 and 0.09, the number that is nearest to the value of $\sqrt{0.9}$ is,

- $0.09$
- $0.3$
- $0.03$
- $0.9$

**Solution 5: Problem analysis and solving by Approximation of square root of a decimal number**

First find the approximate value of $\sqrt{0.9}$.

As first trial, evaluate first $(0.9)^2=0.81$. Falls much below target $0.9$. Next try $(0.95)^2=0.9025$. Just exceeds $0.9$. Approximate value of $\sqrt{0.9}$ can be taken as,

$\sqrt{0.9}=0.94$

Among the four choice values then $0.9$ will be closest to $\sqrt{0.9}$.

**Answer:** Option d: $0.9$.

*Key concepts used:* **Approximate square root estimation of a decimal in square root -- Solving in mind****.**

**Q6.** Which of the following statement(s) is/are TRUE?

- $\sqrt{144}\times{\sqrt{36}} \lt \sqrt[3]{125}\times{\sqrt{121}}$
- $\sqrt{324}+\sqrt{49} \lt \sqrt[3]{216}\times{\sqrt{9}}$

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

**Solution 6: Problem analysis and solution by squares and cubes of common integers and inequality analysis**

First evaluate the four square roots of the first inequality,

$\sqrt{144}=12$,

$\sqrt{36}=6$,

$\sqrt[3]{125}=5$, and

$\sqrt{121}=11$.

$\text{LHS of the first inequality}= 12\times{6}=72$, and

$\text{RHS of the first inequality}=5\times{11}=55 \lt \text{LHS of the first inequality}$.

So the **first inequality is FALSE.** *Only option b or d may be the answer.*

Now evaluate the four square roots of the second inequality,

$\sqrt{324}=18$,

$\sqrt{49}=7$,

$\sqrt[3]{216}=6$, and,

$\sqrt{9}=3$.

$\text{LHS of the second inequality}= 18+7=25$, and

$\text{RHS of the second inequality}=6\times{3}=18 \lt \text{LHS of the first inequality}$.

**Again this inequality is FALSE.**

*Neither I nor II is TRUE.*

**Answer:** Option d : Neither I nor II.

**Key concepts used:** *Squares and cubes of common integers -- Inequality analysis -- Solving in mind.*

**Q7.** If $\displaystyle\frac{(x-\sqrt{24})(\sqrt{75}+\sqrt{50})}{\sqrt{75}-\sqrt{50}}=1$, then the value of $x$ is,

- $3\sqrt{5}$
- $5$
- $\sqrt{5}$
- $2\sqrt{5}$

**Solution 7: Problem analysis and Solving by Pattern identification, Target simplification by Surd term factoring, Surd rationalization**

In this case principle of **simplify the target expression first** is modified for the LHS to,

Before carrying out the product or any other involved operation on the large surd terms, simplify the target expression by simplifying the surds terms themeselves.

**Identify common factor of** $5=\sqrt{25}$ in the denominator and numerator to cancel it out of the terms. The simplified target is,

$\displaystyle\frac{(x-\sqrt{24})(\sqrt{3}+\sqrt{2})}{\sqrt{3}-\sqrt{2}}=1$,

Or, $(x-\sqrt{24})=\displaystyle\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$, factors transposed to the RHS to keep $x$ separate for its easy evaluation

Or, $x-2\sqrt{6}=(\sqrt{3}-\sqrt{2})^2$, denominator rationalized by multiplying and dividing by $(\sqrt{3}-\sqrt{2})$

Or, $x-2\sqrt{6}=5-2\sqrt{6}$

Or, $x=5$.

Solved quite easily in mind.

**Answer:** Option b: 5.

** Key concepts used:** ** Surd rationalization in stages **--

**Surd term factoring -- Avoiding product of surd with $x$, transposition of the factors to the RHS -- Surd arithmetic***--*Solving in mind.**Q8.** The value of $\displaystyle\frac{1}{\sqrt{2}+1}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}+\displaystyle\frac{1}{\sqrt{4}+\sqrt{3}}+.....+\displaystyle\frac{1}{\sqrt{100}+\sqrt{99}}$ is,

- $1$
- $\sqrt{99}$
- $9$
- $\sqrt{99}-1$

** Solution 8: Problem analysis and solving by identifying and using Simplified surd rationalization result pattern**

**Identify the pattern** that the difference of squares of the two terms in each denominator of the terms in the sum is 1.

**Result:** When you rationalize each term, the **denominator is cleanly eliminated.**

Rationalizing all the terms in the sum you get,

$\displaystyle\frac{1}{\sqrt{2}+1}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}+\displaystyle\frac{1}{\sqrt{4}+\sqrt{3}}+.....+\displaystyle\frac{1}{\sqrt{100}+\sqrt{99}}$

$=(\sqrt{2}-1)+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})+.....+(\sqrt{100}-\sqrt{99})$.

Now **identify the second pattern** that the $(-\sqrt{2})$ of the second term cancels out $(\sqrt{2})$ of the first term leaving 1.

The $(-\sqrt{3})$ of the third term cancel out $(\sqrt{3})$ of the second term, thus eliminating both the elements of the second term.

This pattern of cancellation goes on to finally $(-\sqrt{99})$ of the last term cancelling out $(\sqrt{99})$ of the last but one term.

Finally, only $(-1)$ of the first term and, $\sqrt{100}=10$ of the last term are left.

Result is, $10-1=9$.

**Answer:** Option c: 9.

**Key concepts used:** **Simplified surd rationalization pattern identification -- Rationalization of Surds -- Identification of Pattern of term cancellation in the whole series -- Solving in mind.**

**Q9.** If $1^3+2^3+3^3....+10^3=3025$, then the value of $2^3+4^3+....+20^3$ is,

- 7590
- 5060
- 12100
- 24200

**Solution 9: Problem analysis and Solving by term factoring and product rule of equal power terms in a series**

Identify the key pattern that if you take $2^3$ out of each term in the series it is converted simply to,

$2^3+4^3+....+20^3$

$=2^3\times{(1^3+2^3+....+10^3)}$

$=2^3\times{3025}$

$=24200$.

This happens because of the product rule of indices.

$(xy)^p=x^p\times{y^p}$.

**Answer:** Option d: 24200.

**Key concepts used:** * Product rule for variables of equal power *--

**Solving in mind.****Hint:** To get the answer, first eliminate the first two options as $8\times(10^3)=8000$ is larger than both.

Next take the second largest term in the sum of cubes,

$8\times{(1000+729)}=8000+5600+240-8=13832 \gt 12100$.

No need to evaluate the whole sum of cubes.

**Q10.** $\left(\displaystyle\frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}}+\displaystyle\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}\right)$ simplifies to,

- $\sqrt{5}-\sqrt{6}$
- $\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}-3\sqrt{6}$

**Solution 10: Problem analysis and Solution by simple cross-multiplication forming numerator**

Denominator simplifies to,

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

In the numerator you have eight terms,

$(\sqrt{5}-\sqrt{3})(1+\sqrt{2})+(\sqrt{5}+\sqrt{3})(1-\sqrt{2})$

$=(\sqrt{10}-\sqrt{10})+2\sqrt{5}+(-\sqrt{3}+\sqrt{3})-2\sqrt{6}$

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

As denominator already is evaluated as 2, the final result is,

$\sqrt{5}-\sqrt{6}$.

**Answer: **Option a: $\sqrt{5}-\sqrt{6}$.

Observing the similarities in the numerator and denominator terms and the opposite signs of the terms, with a little care you can easily solve the problem in mind.

**Key concepts used:** **Surd arithmetic -- Simplifying elimination by cross-multiplication -- Solving in mind.**

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

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