73rd SSC CGL level Solution Set, 7th on topic Surds and Indices
This is the 73rd solution set of 10 practice problem exercise for SSC CGL exam and 7th on topic Surds and Indices. Students must complete the corresponding question set in prescribed time first and then only refer to this solution set for extracting maximum benefits from this resource.
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73rd solution set—10 problems for SSC CGL exam: 7th on topic Surds and Indices - time 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
We will rationalize both the terms,
$\displaystyle\frac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}+\displaystyle\frac{\sqrt{12}}{\sqrt{3}-\sqrt{2}}$
$=\displaystyle\frac{(3\sqrt{2}-2\sqrt{3})^2}{18-12}+\sqrt{12}(\sqrt{3}+\sqrt{2})$
$=\displaystyle\frac{1}{6}\left(18+12-12\sqrt{6}\right)+6+2\sqrt{6}$
$=5-2\sqrt{6}+6+2\sqrt{6}$
$=11$.
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 within a minute.
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
The pattern we immediately discover in all four surds in the denominators as the presence of square terms, 16, 16, 9 and 9. We will take the square terms out of the square root and simplifiy. This is use of surd term factoring.
$\displaystyle\frac{\sqrt{5}+\sqrt{3}}{\sqrt{80}+\sqrt{48}-\sqrt{45}-\sqrt{27}}$
$=\displaystyle\frac{\sqrt{5}+\sqrt{3}}{4\sqrt{5}+4\sqrt{3}-3\sqrt{5}-3\sqrt{3}}$
$=\displaystyle\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$
$=1$.
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
By targetted surd term factoring, first we will eliminate $\sqrt{6}$ from the first term and $\sqrt{2}$ from the third term as well as rationalize the second term denominator,
$E=\displaystyle\frac{6}{2\sqrt{3}-\sqrt{6}}+\displaystyle\frac{\sqrt{6}}{\sqrt{3}+\sqrt{2}}-\displaystyle\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}$
$=\displaystyle\frac{\sqrt{6}}{\sqrt{2}-1}+\sqrt{6}(\sqrt{3}-\sqrt{2})-\displaystyle\frac{2\sqrt{6}}{\sqrt{3}-1}$
Now we will rationalize the simple denominators of the first and third term,
$E=\sqrt{6}(\sqrt{2}+1+\sqrt{3}-\sqrt{2}-\sqrt{3}-1)$
$=0$.
Simplification in a few steps depended upon the suitable surd term factoring for canceling out surd factors between the numerators and denominators as well as the strategy of first simplifying the denominator and then only rationalize.
Answer: Option c: $0$.
Key concepts used: Surd term factoring -- Denominator simplification first and then 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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