Simplification MCQ Quiz - Objective Question with Answer for Simplification - Download Free PDF

Given:

(125)^x = 3125

Formula Used:

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Calculation:

We know that 125 = 5³, so we can rewrite the equation:

(5³)^x = 3125

Using the power of a power property, (a^m)ⁿ = a^m×n:

⇒ 5^3x = 3125

We also know that 3125 = 5⁵, so we can rewrite the equation:

⇒ 5^3x = 5⁵

Since the bases are the same, we can equate the exponents:

⇒ 3x = 5

⇒ x = 5/3

The correct answer is option 4.

Concept Used:

Calculation:

⇒ \(\rm 864÷ \left\{\frac{3}{4}\left[\frac{16}{15}\right]-\frac{2}{3}\right\} \)

⇒ 864 ÷ {3/4 × 16/15 - 2/3}

⇒ 864 ÷ {4/5 - 2/3}

⇒ 864 ÷ {\(\frac{4 × 3\ - \ 2 × 5}{5 × 3}\)}

⇒ 864 ÷ {\(\frac{12 \ - \ 10}{15}\)}

⇒ 864 ÷ 2/15

⇒ 864 × 15/2

⇒ 6480

∴ The simplified value is 6480

Calculation

⇒ \(2\frac{5}{8} \times 2\frac{1}{7} \div 4\frac{8}{5} \times5\frac{3}{5}=?\)

⇒ 21/8 × 15/7 ÷ 28/5 × 28/5

⇒ 21/8 × 15/7 × 5/28 × 28/5 

⇒ 21/8 × 15/7 × 1

⇒ (45/8) × 1

= \(5\frac{5}{8}\)

∴ The answer is \(5\frac{5}{8}\)

Given:

35.17 + 21.53 + 37.61 - 42.86a+b+c−d=3×?" id="MathJax-Element-265-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-17-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-75-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-42-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-45-Frame" role="presentation" style="position: relative;" tabindex="0"> $a + b + c - d = 3 × ?$

Calculation:

35.17 + 21.53 + 37.61 - 42.86

⇒ 35.17 + 21.53 = 56.70

⇒ 56.70 + 37.61 = 94.31

⇒ 94.31 - 42.86 = 51.45

Expression: 3 × ? = 51.45

⇒ ? = 51.45 / 3

⇒ ? = 17.15

The correct answer is option 1.

Given:

(√13 − 1)/(√13 + 1) = a + b√13

Formula used:

Rationalize denominator: (a − b)/(a + b) = [(a − b)(a − b)] / [(a + b)(a − b)]

Use: (a + b)(a − b) = a² − b²

Calculation:

(√13 − 1)/(√13 + 1)

⇒ Multiply numerator and denominator by (√13 − 1)

⇒ [(√13 − 1) × (√13 − 1)] / [(√13 + 1) × (√13 − 1)]

⇒ (13 − 2√13 + 1) / (13 − 1)

⇒ (14 − 2√13) / 12

⇒ 14/12 − (2√13)/12

⇒ 7/6 − (√13)/6

∴ a = 7/6 and b = −1/6

Concebt used

a.b̅ = a.bbbbbb

a.0b̅ = a.0bbbb

Calculation

0.7 = 0.700000 ̇....

\(0.\bar7 = 0.77777 \ldots\)

\(0.0\bar7 = 0.077777 \ldots\)

\(0.\overline {07} = 0.070707 \ldots\)

Solution:

\(12\frac{1}{2} + 12\frac{1}{3} + 12\frac{1}{6}\)

= 25/2 + 37/3 + 73/6

= (75 + 74 + 73)/6

= 222/6

= 37

Shortcut Trick

\(12\frac{1}{2} + 12\frac{1}{3} + 12\frac{1}{6}\)

= 12 + 12 + 12 + (1/2 + 1/3 + 1/6)

= 36 + 1 = 37

Solution:

We have to follow the BODMAS rule

Calculation:

\(? = \sqrt[5]{{{{\left( {243} \right)}^2}}}\)

\(⇒ ? = (243)^{\frac{2}{5}}\)

\(⇒ ? = (3 × 3 × 3 × 3 × 3)^{2⁄5}\)

\(⇒ ? = (3^5)^{2⁄5}\)

⇒ ? = 3²

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

Given expression is:

\(\sqrt {8\; + \;2\sqrt {15} \;} \)

⇒  \(\sqrt {5\; + \;3\; + \;2\times \sqrt 5 \times \sqrt 3 \;} \)

⇒  \(\sqrt {{{(\sqrt 5 )}^2}\; + \;{{\left( {\sqrt 3 } \right)}^2}\; + \;2 \times \sqrt 5 \times \sqrt 3 \;} \)

⇒  \(\sqrt {{{\left( {\;\sqrt 5 \; + \;\sqrt 3 \;} \right)}^2}\;} \)

⇒  \(\sqrt 5 + \sqrt 3 \)

\(\sqrt {{{\left( {0.65} \right)}^2} - {{\left( {0.16} \right)}^2}} \)

Since,

a2 - b2 = (a - b) ( a + b)

\(\begin{array}{l} \Rightarrow \sqrt {\left( {0.65 + 0.16} \right)\left( {0.65 - 0.16} \right)} \\ \Rightarrow \sqrt {\left( {0.81} \right)\left( {0.49} \right)} \\ \Rightarrow \sqrt {\left( {0.9} \right)\left( {0.9} \right) \times \left( {0.7} \right)\left( {0.7} \right)} \end{array}\)

⇒ 0.9 × 0.7 = 0.63

Concept:

We can find √x using the factorisation method.

Calculation:

√[(10 + √25) (12 - √49)]

⇒ √[(10 + 5)(12 – 7)]

⇒ √(15 × 5)

⇒ √(3 × 5 × 5)

⇒ 5√3

Given,

2³ × 3⁴ × 1080 ÷ 15 = 6^x

⇒ 2³ × 3⁴ × 72 = 6^x

⇒ 2³ × 3⁴ × (2 × 6²) = 6^x

⇒ 2⁴ × 3⁴ × 6² = 6^x

⇒ (2 × 3)⁴ × 6² = 6^x           [∵ x^m × y^m = (xy)^m]

⇒ 6⁴ × 6² = 6^x

⇒ 6^{(4 + 2)} = 6^x

⇒ x = 6

Given:

√3n = 729

Formulas used:

(x^a)^b = x^ab

If x^a = x^b then a = b 

Calculation:

√3n = 729

⇒ √3n = (3²)³

⇒ (3n)^1/2 = (32)3

⇒ (3n)1/2 = 3⁶

⇒ n/2 = 6 

∴  n = 12 

Given expression,

(81.84 + 118.16) ÷ 5³ = 1.2 × 2 + ?

⇒ 200 ÷ 5³ = 1.2 × 2 + ?

⇒ 200 ÷ 125 = 1.2 × 2 +?

⇒ 1.6 = 2.4 + ?

⇒ ? = -0.8