Robert Bosch Surds and Indices Quiz 4

Explanation:

2^{60} = (2^{5})^{12} = 32^{12}

5^{24} = (5^{2})^{12} = 25^{12}

\therefore 2^{60}> 5^{24}

3^{48} = (3^{4})^{12} = 81^{12}

\therefore 3^{48}  >2^{60}

4^{36} = (4^{3})^{12} = 64^{12}

\therefore 3^{48} is the largest number.

 

Explanation:

3\sqrt{2}  = 3 \times 1.4 = 4.2

3\sqrt{7} = 3 \times 2.6 = 7.8

6\sqrt{5} = 6 \times 2.2 = 13.2

2\sqrt{20}= 2 \times 4.5 = 9

 

Explanation:

\sqrt{0.16}. = 0.4; (0.16)^{2}= 0.0256

Clearly,

0.0256 < 0.04 < 0.16 < \sqrt{0.16}

Explanation:

2^{250} = (2^{5} ) ^{50} = (32)^{50}

3^{ 150} = (3^{3} )^{50} = (27)^{50}

5 ^{100} = (5^{2 })^{ 50} = (25)^{50}

4^{ 200} = (4^{4 }) ^{50} = (256)^{50}

\therefore The smallest number = (5)^{100}

Explanation:

3\sqrt{ 3} \times \sqrt{3} = 3 \times 3= 9

\therefore Required rationalising factor is 3

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Explanation:

 

 6^{ 333} \times 7^{222} \times 8^{111}

 \therefore (2 x3)^{ 333} \times 7^{222} \times 8^{111} 

  \therefore 2^{ 333} \times 3^{333} \times7^{222} \times 8^{111}

  \therefore 2^{ 666}\times 3^{333}\times 7^{222}

  \therefore Number of prime factors = 666 + 333 + 222 = 1221 

Explanation:

 27^{2x – 1} = (243)^{3} 

=>  (3^{3} )^{ 2x–1} = (35 )^{ 3} 

=>   (3)^{3(2x–1)} = (3)^{5 × 3} 

=>  36^{(2x – 1)} = 5 \times 3 or 2x – 1 = 5 

  \therefore  x = 3 

Explanation:

3^{x+8} = 3^{3(2x+1)} 

=> x + 8 = 6x + 3 

=> 5x = 5 

\therefore  x = 1 

 

Explanation:

(36)^{\frac{1}{6}} = (6^{2])^{\frac{1}{6}} 

=  (6)^{\frac{2}{6}} =(6)^{\frac{1}{3}} =\sqrt[3]{6}