SSC CGL level Solution Set 61, Fractions indices and surds 5

Comparison of surds, surds and indices solution set 61

Learn comparison of surds and surds simplification in SSC CGL solutions 61 and how to solve 10 surds and indices questions in 12 minutes.

Contents are,

1. Solution to Comparison of surds questions
2. Solution to Surds and indices questions.
3. Method to compare surds.

If you have not taken the companion test yet, first take the test on SSC CGL level Question set 61 on fractions indices surds 5 and then go through the solutions for gaining maximum benefit.

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10 questions on comparison of surds, surds and indices SSC CGL solution set 61 - time to solve was 12 mins

Problem 1.

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

1. 34
2. 37
3. 36
4. 35

Solution 1: Problem analysis and solving

We will break up the base terms under powers into product of prime factors, count the numbers of primes factors in each, multiply each such number with the respective powers of the bases and add up to get the required total number of prime factors.

The given expression,

$4^{10}\times{16^2}\times{7^3}\times{11}\times{10^2}$.

The first product term gives,

$4^{10}=(2\times{2})^{10}\rightarrow\text{ 20 prime factors}$.

The second product term gives,

$16^2=(2\times{2}\times{2}\times{2})^{2}\rightarrow\text{ 8 prime factors}$.

The third product term gives,

$7^3\rightarrow\text{ 3 prime factors}$.

The fourth product term gives,

$11\rightarrow\text{ 1 prime factor}$.

The fifth product term gives,

$10^2=(2\times{5})^{2}\rightarrow\text{ 4 prime factors}$.

The total number of prime factors is then,

$20+8+3+1+4=36\text{ prime factors}$.

The total could be arrived at mentally in quick time.

Answer: Option c: 36.

Key concepts used: Basic indices concepts -- factors and multiples concepts.

Problem 2.

Arrange the following in descending order,

$\sqrt[3]{4}$, $\sqrt{2}$, $\sqrt[6]{3}$, $\sqrt[4]{5}$.

1. $\sqrt[6]{3} \gt \sqrt[4]{5} \gt \sqrt[3]{4} \gt \sqrt{2}$
2. $\sqrt{2} \gt \sqrt[6]{3} \gt \sqrt[3]{4} \gt \sqrt[4]{5}$
3. $\sqrt[3]{4} \gt \sqrt[4]{5} \gt \sqrt{2} \gt \sqrt[6]{3}$
4. $\sqrt[4]{5} \gt \sqrt[3]{4} \gt \sqrt[6]{3} \gt \sqrt{2}$

Solution 2: Problem analysis

Each of the four terms has a base and a fraction power because each is some type of root.

To compare the terms we may consider using the technique of equalizing the bases or the powers. But the problem given has different prime bases so that bases can't be equalized in this case.

When we consider the choice of equalizing the powers, only one option turns out to be the possible path to solution. The target power to which we need to equalize the term powers must be 1 with bases (transformed if required)  pure numeric values. This is possible by raising the each of the root terms to the power of 12 which is the LCM of the fraction power denominators, 3, 2, 6, and 4.

The assumption here is, when we raise two terms by same positive power (negative power reverses comparison result), their relative value comparison result doesn't change, the smaller one remains smaller only. This is by Inequality in indices concepts.

If, $\sqrt{a} \gt \sqrt{b}$, it will be true that,

$\left(\sqrt{a}\right)^2 \gt \left(\sqrt{b}\right)^2$, we have taken power 2 as an example, it could have been any positive power.

The main obejective will then be,

Bringing out all the terms from under roots to pure numeric values.

To achieve this objective, we will have to raise all the terms to a power that is the LCM of the denominators of the root powers.

This is Comparison of values under roots technique and is a special case of Base equalization technique.

Solution 2: Problem solving by Base equalization technique

The four given terms are,

$\sqrt[3]{4}$, $\sqrt{2}$, $\sqrt[6]{3}$, $\sqrt[4]{5}$.

Raising each term to its 12th power the terms are transformed to,

$4^4$, $2^6$, $3^2$, $5^3$,

Or, $256$, $64$, $9$, $125$.

So the descending order arrangement will be,

$256$, $125$, $64$, $9$,

Or, $\sqrt[3]{4}$, $\sqrt[4]{5}$, $\sqrt{2}$, $\sqrt[6]{3}$.

Answer: Option c: $\sqrt[3]{4}$, $\sqrt[4]{5}$, $\sqrt{2}$, $\sqrt[6]{3}$.

Key concepts used: Basic indices concepts -- Inequality in indices concept -- Base equalization technique -- Equalization of root powers -- Comparison of values under roots.

Problem 3.

If $\sqrt{15}=3.88$, then the value of $\sqrt{\displaystyle\frac{5}{3}}$ is,

1. $1.29$
2. $1.295$
3. $1.29\overline{3}$
4. $1.2934$

Solution 3: Problem analysis and solving

$\sqrt{15}=3.88$,

Or, $\displaystyle\frac{\sqrt{15}}{\sqrt{9}}=\frac{3.88}{3}$,

Or, $\sqrt{\displaystyle\frac{5}{3}}=1.29\overline{3}$.

Answer: Option c: $1.29\overline{3}$.

Key concepts used: Key pattern identification -- End state analysis.

Problem 4.

The simplified value of $(\sqrt{3}+1)(10+\sqrt{12})(\sqrt{12}-2)(5-\sqrt{3})$ is equal to,

1. 132
2. 16
3. 88
4. 176

Solution 4: Problem analysis and solving execution

The given expression,

$E=(\sqrt{3}+1)(10+\sqrt{12})(\sqrt{12}-2)(5-\sqrt{3})$

$=\left[2(\sqrt{3}+1)(\sqrt{3}-1)\right](10+\sqrt{12})(5-\sqrt{3})$

$=4\left[2(5+\sqrt{3})(5-\sqrt{3})\right]$

$=8(25-3)$

$=176$.

Answer: Option d: 176.

Key concepts used: Surd simplification -- Surd factorization -- basic algebraic concepts.

Problem 5.

$\displaystyle\frac{12}{3+\sqrt{5}+2\sqrt{2}}$ is equal to,

1. $1-\sqrt{5}-\sqrt{2}+\sqrt{10}$
2. $1+\sqrt{5}-\sqrt{2}+\sqrt{10}$
3. $1+\sqrt{5}+\sqrt{2}-\sqrt{10}$
4. $1-\sqrt{5}+\sqrt{2}+\sqrt{10}$

Solution 5: Problem analysis

The choice values having no denominator, the main goal will be to eliminate the denominator in the target expression.

In a surd fraction, rationalization converts the surd denominator to an integer and effectively eliminates it by applying $(a+b)(a-b)=a^2-b^2$, where one of $a$ or $b$ is a surd.

Ratonalization is straightforward in case of a two term surd denominator expression.

In this case though the denominator is a three term surd expression and it calls for special measure.

The only way forward in this case is thus to rationalize, but not once. We need to rationalize the denominator twice.

In the first step, the two surd terms are to be split into the two rationalizing factors so that one of those is transformed to an integer.

Let us show how.

Solution 5: Problem solving execution

The target expression,

$E=\displaystyle\frac{12}{3+\sqrt{5}+2\sqrt{2}}$

$=\displaystyle\frac{12[(3+\sqrt{5})-2\sqrt{2}]}{(3+\sqrt{5})^2-8}$, first stage rationalization

$=\displaystyle\frac{12[(3+\sqrt{5})-2\sqrt{2}]}{6+6\sqrt{5}}$

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

$=\displaystyle\frac{1}{2}[(3+\sqrt{5}-2\sqrt{2})(\sqrt{5}-1)]$, second stage rationalization

$=\displaystyle\frac{1}{2}[3\sqrt{5}+5-2\sqrt{10}-3-\sqrt{5}+2\sqrt{2}]$

$=\displaystyle\frac{1}{2}[2+2\sqrt{5}+2\sqrt{2}-2\sqrt{10}]$

$=1+\sqrt{5}+\sqrt{2}-\sqrt{10}$.

Answer: Option c: $1+\sqrt{5}+\sqrt{2}-\sqrt{10}$.

Key concepts used: Deductive reasoning -- Double rationalization -- Goal directed rationalization term selection -- Surd simplification.

Problem 6.

The value of $\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{1}{5-\displaystyle\frac{1}{5}}}}}}$ is,

1. 1
2. 2
3. 3
4. 4

Solution 6: Problem analysis and solving execution

The target expression needs to be simplified by starting from the lowermost fraction expression simplification in the continued fraction.

The target expression,

$E=\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{1}{5-\displaystyle\frac{1}{5}}}}}}$

$=\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{1}{\displaystyle\frac{24}{5}}}}}}$

$=\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{5}{24}}}}}$

$=\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{2+\displaystyle\frac{1}{\displaystyle\frac{53}{24}}}}}$

$=\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{2+\displaystyle\frac{24}{53}}}}$

$=\sqrt{\displaystyle\frac{1+\displaystyle\frac{8}{7}-\displaystyle\frac{1}{4}}{4+\displaystyle\frac{1}{2}+\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{\displaystyle\frac{130}{53}}}}$, applying sum and subtraction of integer portions of mixed fractions

$=\sqrt{\displaystyle\frac{1+\displaystyle\frac{25}{28}}{4+\displaystyle\frac{9}{14}}\div{\displaystyle\frac{53}{130}}}$

$=\sqrt{\displaystyle\frac{\displaystyle\frac{53}{28}}{\displaystyle\frac{65}{14}}\times{\displaystyle\frac{130}{53}}}$

$=\sqrt{\displaystyle\frac{53}{130}\times{\displaystyle\frac{130}{53}}}$

$=1$.

Answer: Option a: 1.

Key concepts used: Continued fraction -- fraction simplifcation -- efficient simplification -- problem breakdown technique.

Note: While solving this type of problem in actual test, we don't write down all the steps, but only jot down the intermediate results that need remembering. This method speeds up the step by step simplification process even for this type of awkward looking problem. The important point here is to carry out one step at a time and note down just the single result of the step.

Better still is to evaluate the three parts individually, the numerator and the denominator of the first compound fraction and the continued fraction. That is problem breakdown into three parts.

Problem 7.

Which is the greatest among $(\sqrt{19}-\sqrt{17})$, $(\sqrt{13}-\sqrt{11})$, $(\sqrt{7}-\sqrt{5})$, and $(\sqrt{5}-\sqrt{3})$?

1. $(\sqrt{5}-\sqrt{3})$
2. $(\sqrt{7}-\sqrt{5})$
3. $(\sqrt{19}-\sqrt{17})$
4. $(\sqrt{13}-\sqrt{11})$

Solution 7: Problem analysis and solving execution

To find the greatest, smallest among a set of expressions or to order them in descending or ascending sequence, the main action to carry out repeatedly is to compare two of the expressions taken together.

Applying equal difference surd comparison concept on the first two expressions we conclude,

$(\sqrt{19}-\sqrt{17}) \lt (\sqrt{13}-\sqrt{11})$.

This is because the difference between each two terms of the two expressions being 2, and $19 \gt 13$, the first expression will be smaller.

Taking the last two expressions similarly,

$(\sqrt{7}-\sqrt{5})\lt (\sqrt{5}-\sqrt{3})$.

So we need only to compare the two larger expressions,

$(\sqrt{13}-\sqrt{11})$, $(\sqrt{5}-\sqrt{3})$.

For these two expressions also equal difference between two terms is 2 and so the expression with smaller value under square root, that is, $(\sqrt{5}-\sqrt{3})$ will be the largest.

Answer: Option a: $(\sqrt{5}-\sqrt{3})$.

Key concepts used: Surd expression series comprison -- Key pattern identification -- Equal difference surd comparison.

Alternative faster logic

Alternatively examining all the four terms we find the difference between two values under square roots for each expression is equal to 2. Among all four expressions, $(\sqrt{5}-\sqrt{3})$ having the smallest value of the positive term under square roots $\sqrt{5}$, this expression will be the largest by the equal difference surd terms comparison concept.

We need just to identify the key pattern and apply the powerful equal difference surd comparison concept.

To know more about the equal difference surd comparison concept and its mechanism, refer to our earlier conceptual solution session on SSC CGL level Solution set 60 on fractions indices and surds 4.

Problem 8.

If $\sqrt{3}=1.732$ what is the value of $\displaystyle\frac{4+3\sqrt{3}}{\sqrt{7+4\sqrt{3}}}$?

1. 0.464
2. 0.023
3. 3.023
4. 2.464

Solution 8: Problem analysis and solving execution

We need to simplify the target expression first and then substitute the given value,

$E=\displaystyle\frac{4+3\sqrt{3}}{\sqrt{7+4\sqrt{3}}}$

$=\displaystyle\frac{4+3\sqrt{3}}{\sqrt{(2+\sqrt{3})^2}}$, by surd expression transformation to a square expression

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

$=\displaystyle\frac{(4+3\sqrt{3})(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$, rationalizing the denominator

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

$=8-4\sqrt{3}+6\sqrt{3}-9$

$=2\sqrt{3}-1$

$=3.464-1$

$=2.464$

Answer: Option d: 2.464.

Key concepts used: Surd expression transformation to square form -- Surd rationalization -- Surd simplification.

Problem 9.

The simplified value of $\left[\sqrt[3]{\sqrt[6]{5^9}}\right]^4\left[\sqrt[3]{\sqrt[6]{5^9}}\right]^4$ is,

1. $5^4$
2. $5^8$
3. $5^{12}$
4. $5^2$

Problem analysis and solving execution

The powers need to be represented in fraction form instead of root form to simplifiy the effect of power on powers.

The given expression,

$\left[\sqrt[3]{\sqrt[6]{5^9}}\right]^4\left[\sqrt[3]{\sqrt[6]{5^9}}\right]^4$

$=\left[\sqrt[3]{5^{\frac{9}{6}}}\right]^4\left[\sqrt[3]{5^{\frac{9}{6}}}\right]^4$

$=\left[5^{\frac{9}{18}}\right]^4\left[5^{\frac{9}{18}}\right]^4$

$=\left[5^{\frac{1}{2}}\right]^4\left[5^{\frac{1}{2}}\right]^4$

$=5^2\times{5^2}$

$=5^4$.

Answer. Option a: $5^4$.

Key concepts used: Basic indices concepts -- Surd indices concepts.

Problem 10.

The smallest of $\sqrt{8}+\sqrt{5}$, $\sqrt{7}+\sqrt{6}$, $\sqrt{10}+\sqrt{3}$, $\sqrt{11}+\sqrt{2}$ is,

1. $\sqrt{7}+\sqrt{6}$
2. $\sqrt{8}+\sqrt{5}$
3. $\sqrt{10}+\sqrt{3}$
4. $\sqrt{11}+\sqrt{2}$

Solution 10: Problem analysis and solving

Examining the four surd expressions we find that the expressions are neither in sutractive form nor the term differences are equal. But still we find possibility of comparing the first two expressions and the last two expressions by transposing terms suitably.

Let us then compare the first two expressions,

$\sqrt{8}+\sqrt{5}$, $\sqrt{7}+\sqrt{6}$.

Observing that if we subtract $\sqrt{7}$ from $\sqrt{8}$ and again subtract $\sqrt{5}$ from $\sqrt{6}$ we will have a conclusive relation by the equal difference surd comparison concept,

$\sqrt{8}-\sqrt{7} \lt \sqrt{6}-\sqrt{5}$,

Or, $\sqrt{8}+\sqrt{5} \lt \sqrt{7}+\sqrt{6}$, by basic inequality arithmetic.

Similarly we will have,

$\sqrt{10}+\sqrt{3}\gt \sqrt{11}+\sqrt{2}$.

Now we have to compare the two smaller valued expressions,

$\sqrt{8}+\sqrt{5}$ and $\sqrt{11}+\sqrt{2}$.

Again we find the possibility of applying the equal difference surd comparison concept by transposing terms,

$\sqrt{11}-\sqrt{8} \lt \sqrt{5}-\sqrt{2}$,

Or, $\sqrt{11}+\sqrt{2} \lt \sqrt{8}+\sqrt{5}$.

Thus, $\sqrt{11}+\sqrt{2}$ will be the smallest among the four.

Answer: Option d: $\sqrt{11}+\sqrt{2}$.

Key concepts used: Key pattern identification -- Inequality analysis by transposition -- Inequality arithmetic -- Equal difference surd comparison concept.

More efficient solution by applying the rich concept of Equal sum surd comparison concept

Let us state this rich surd expression comparison concept that is based on the equal difference surd comparison concept.

If in two surd expressions,

$\sqrt{a}+\sqrt{b}$ and $\sqrt{c}+\sqrt{d}$

the sum of the terms under roots are equal,

$a+b=c+d$,

and

$a \gt c$, where $a$ and $c$ are the larger of the two term values in the two expressions respectively

then,

$\sqrt{a}+\sqrt{b} \lt \sqrt{c}+\sqrt{d}$.

Mechanism of the rich concept of Equal sum surd comparison concept

The two expressions to be compared are,

$\sqrt{a}+\sqrt{b}$ and $\sqrt{c}+\sqrt{d}$.

Here,

$a+b=c+d$,

Or, $a-d=c-b$.

As $a \gt c$, where $a$ and $c$ are the larger terms in each expression, by equal difference surd comparison concept,

$\sqrt{a}-\sqrt{d} \lt \sqrt{c}-\sqrt{b}$,

Or, $\sqrt{a}+\sqrt{b} \lt \sqrt{c}+\sqrt{d}$, by inequality arithmetic.

Alternate solution 10: By applying equal sum surd comparison concept

Sum of the surd terms under roots for each of the four given expressions is equal to 13. The larger values in the expressions are, 7, 8, 10 and 11. So the expression with the largest of the larger values, $\sqrt{11}+\sqrt{2}$ will be the smallest.

This is by far the fastest solution because we have applied the second level rich surd expression comparison concept derived from first level rich concept.

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Suresolv Fractions, Surds and Indices Reading and Practice Guide for SSC CHSL, SSC CGL, SSC CGL Tier II and Other Competitive exams.

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