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GIVEN,

⇒ 1/3(12x/5 - 1/2) + 6/5 = 7/6

⇒ 4x/5 - 1/6 + 6/5 = 7/6

⇒ 4x/5 = 7/6 + 1/6 - 6/5

⇒ 4x/5 = 2/15

⇒ X = (2/15)/(4/5)

∴ x = 1/6

LAWS of Indices:

1. a^m × aⁿ = a^{{m + n}}

2. a^m ÷ aⁿ = a^{{m - n}}

3. (a^m )ⁿ = a^mn

4. (a)^-m = 1/a^m

5. (a)^m/n = ⁿ√a^m

6. (a)⁰ = 1

⇒ 5¹² × 125 ÷ 15625 = 3125 × 25^?

⇒ 5¹² × 5³ ÷ 5⁶ = 5⁵ × (5²)^?

⇒ 5^{12 + 3 - 6} = 5^{5 + 2?}

⇒ 5⁹ = 5^{5 + 2?}

⇒ 9 = 5 + 2?

⇒ ? = 4/2

GIVEN,

x = √3-1

Put the value of x in given equation

⇒ x² + 2√3 = (√3 - 1)² + 2√3

Follow BODMAS RULE to solve this question, as per the order given below,

Step - 1 - Parts of an equation enclosed in 'Brackets' must be solved FIRST,

Step - 2 - Any mathematical 'Of' or 'Exponent' must be solved next,

Step - 3 - Next, the parts of the equation that contain 'Division' and 'Multiplication' are CALCULATED,

Step - 4 - Last but not LEAST, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated.

x = 172 + [(132 ÷ 6 × 5 + 78 × 4 ÷ 3) - {125 ÷ (160 ÷ 32) + 7}]

⇒ x = 172 + [(22 × 5 + 312 ÷ 3) - {125 ÷ 5 + 7}]

⇒ x = 172 + [(110 + 104) - {25 + 7}]

⇒ x = 172 + [214 - 32]

⇒ x = 172 + 182

⇒ x = 354

⇒ (25/7) × (13/5) × (7/6) ÷ (19/7) = ?

$(\Rightarrow \FRAC{{25}}{7} \TIMES \frac{{13}}{5} \times \frac{7}{6} \times \frac{7}{{19}} = ?)$

⇒ (25 × 13 × 7) ÷ (5 × 6 × 19) = ?

⇒ 2275/570 = ?

125 - 73 + 48 - 137 + 99 = ?

⇒ ? = 52 + 48 - 137 + 99

⇒ ? = 100 - 137 + 99

⇒ ? = -37 + 99

By the BODMAS RULE,

Solving the Brackets,

1875.96 ≈ 1876

500.45 ≈ 500

1876 - 500 = 1376

399.8 ≈ 400

900 + 400 = 1300

Solving the division,

1376 ÷ 1300 ≈ 1.058 ≈ 1

Trick:

No need to do CALCULATIONS. Just observe two things. Firstly, SOMETHING is being subtracted from 78 which means answer would be less than 78. Secondly, we have to check WHETHER the term in the brackets is negative or POSITIVE. It would be negative if1.95 × 9.998 would be greater than 24.98 which is not true. Hence number would be less than 78 and only option 58 is less than 78. Option 1 is the right answer.

Detailed solution:

It can be approximated as 78 – [5 + 3 of (25 – 2 × 10)],

Now using BODMAS rule

Follow BODMAS rule to solve this question, as PER the order given below,

Step-1: Parts of an equation enclosed in 'Brackets' must be solved first, and in the bracket,

Step-2: Any mathematical 'Of' or 'Exponent' must be solved next,

Step-3: Next, the parts of the equation that contain 'Division' and 'Multiplication' are calculated,

Step-4: Last but not least, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated.

⇒ 78 – [5 + 3 of (25 – 20)], (Simplifying ‘multiplication’ 2 × 10 = 20)

⇒ 78 – [5 + 3 of 5], (Simplifying ‘subtraction’ 25 – 20 = 5)

⇒ 78 – [5 + 3 × 5], (Simplifying ‘of’)

⇒ 78 – [5 + 15], (Simplifying ‘multiplication’ 3 × 5 = 15)

⇒ 78 – 20, (Simplifying ‘addition’ 5 + 15 = 20)

⇒ 58, (Simplifying ‘subtraction’ 78 – 20 = 58)

∴ its approximated value is 58

The key is to use the BODMAS Sequence: Brackets Of Division Multiplication Addition Subtraction.

Solving ACCORDINGLY,

$(\begin{ARRAY}{l}\frac{{1\frac{7}{{12}}}}{{3\frac{4}{5}}} \DIV \left( {\frac{5}{6} \times \frac{{12}}{{15}} + \frac{1}{4}} \right) + \frac{5}{7} \div \frac{{11}}{{13}}{\rm{of\;}}\frac{{13}}{7}\\ = \frac{{1\frac{7}{{12}}}}{{3\frac{4}{5}}} \div \left( {\frac{2}{3} + \frac{1}{4}} \right) + \frac{5}{7} \div \frac{{11}}{{13}}{\rm{of\;}}\frac{{13}}{7}\\ = \frac{{1\frac{7}{{12}}}}{{3\frac{4}{5}}} \div \frac{{11}}{{12}} + \frac{5}{7} \div \left( {\frac{{11}}{{13}} \times \frac{{13}}{7}} \right)\\ = \frac{{1\frac{7}{{12}}}}{{3\frac{4}{5}}} \div \frac{{11}}{{12}} + \frac{5}{7} \div \frac{{11}}{7}\\ = \frac{{\frac{{19}}{{12}}}}{{\frac{{19}}{5}}} \div \frac{{11}}{{12}} + \frac{5}{7} \div \frac{{11}}{7}\\ = \frac{5}{{12}} \times \frac{{12}}{{11}} + \frac{5}{7} \times \frac{7}{{11}}\\ = \frac{5}{{11}} + \frac{5}{{11}}\END{array})$

FACTORS of √2 = 1.414

Factors of ?3 = 1.442

Factors of √4 = 2

Factors of ?5 = 1.709

$(\sqrt 5= {5^{\frac{1}{2}}},\;\sqrt[3]{6} = {6^{\frac{1}{3}}},\;\sqrt[4]{7} = {7^{\frac{1}{4}}},\;\sqrt[6]{8} = {8^{\frac{1}{6}}})$

LCM of denominators of exponents = LCM (2, 3, 4, 6) = 12

$({5^{\frac{1}{2} \times \frac{{12}}{{12}}}},\;{6^{\frac{1}{3} \times \frac{{12}}{{12}}}},\;{7^{\frac{1}{4} \times \frac{{12}}{{12}}}},\;{8^{\frac{1}{6} \times \frac{{12}}{{12}}}} = {5^{\frac{6}{{12}}}},\;{6^{\frac{4}{{12}}}},\;{7^{\frac{3}{{12}}}},\;{8^{\frac{2}{{12}}}})$

5⁶ > 6⁴ > 7³ > 8²

∴ √5 is the greatest

⇒ (.4)² + (.04)² + (.004)² = 0.16 + 0.0016 + 0.000016 = 0.161616

Given, Equation is,

$(\BEGIN{array}{l} \SQRT {0.0016} = ?\\ \Rightarrow \sqrt {\FRAC{{16}}{{10000}}} = ?\\ \Rightarrow \frac{4}{{100}} = ? \end{array})$

3/5 = 0.6

6/7 = 0.85

5/11 = 0.45

We can OBSERVE that 0.85 is LARGEST AMONG all.

∴ 6/7 is largest FRACTION among 3/5, 6/7, 5/11.

$({\left( {0.008} \right)^{\FRAC{1}{3}}} - {\left( {0.0016} \right)^{\frac{1}{4}}} \TIMES {5^0}\; + \;{\left( {\frac{4}{7}} \right)^{ - 1}}\; = \;0.2 - 0.2\; + \;\frac{7}{4}\; = \;1.75)$

⇒ 497.01 + 12.87 - 54.80 = ? % of 500

We can write the GIVEN values as -

⇒ 497.01 ≈ 497 and 12.87 ≈ 13 and 54.80 ≈ 55

Then, the EXPRESSION becomes =

⇒ 497 + 13 - 55 = ? % of 500

⇒ 497 + 13 - 55 = (?/100) × 500

⇒ 497 + 13 - 55 = 5 × ?

⇒ 510 - 55 = 5 × ?

⇒ 455 = 5 × ?

⇒ ? = 455/5

Follow BODMAS rule to SOLVE this QUESTION, as per the order given below,

Step–1: PARTS of an equation enclosed in 'Brackets' must be solved first, and in the bracket,

Step–2: Any MATHEMATICAL 'Of' or 'Exponent' must be solved next,

Step–3: Next, the parts of the equation that contain 'Division' and 'Multiplication' are calculated,

Step–4: Last but not least, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated.

1124 + [24 + {23 – (27 + 6 × 9) + 91} + 66.67% of 75] = ?² + 270

⇒ 1124 + [24 + {23 – (27 + 54) + 91} + 2/3 × 75] = ?² + 270

⇒ 1124 + [24 + {23 – 81 + 91} + 2 × 25] = ?² + 270

⇒ 1124 + [24 + 33 + 50] = ?² + 270

⇒ 1124 + 107 = ?² + 270

⇒ ?² = 1231 – 270 = 961

By looking at the options we can SAY that the options B & C can be RULED out as they result in a terminating NON - recurring decimal numbers i.e.,

18/100 = 0.18 & 18/1000 = 0.018

So let us check the other options:

$(\begin{array}{l} 1/55{\rm{\;}} = {\rm{\;}}0.0181818 \ldots .{\rm{\;}} = {\rm{\;}}0.0\overline {18} \\ \therefore {\rm{\;}}0.0\overline {18} \; = \;\frac{1}{{55}} \END{array})$

<P>⇒ LET the original fraction be p/q

⇒ Given, p × (300/q) × 250 = 3/14

Given expression:

$(\frac{{\frac{4}{{15}}}}{{\frac{2}{5}}} = ?)$

$(\Rightarrow ? = \frac{4}{{15}} \div \frac{2}{5} = \frac{4}{{15}} \TIMES \frac{5}{2} = \frac{2}{3})$