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Evaluate without using tables or calculators: sqrt(0.8064 6.048 / 1.008 0.1344) and -4 of [(-4 + -5 / 15) + -3 - 4 / 6] / (84 / -7 + 3 - (-5)).

Mathematics

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Here are the solutions to your problems:

1. Evaluate without using tables or calculators.

0.8064×6.0481.008×0.1344\sqrt{\frac{0.8064 \times 6.048}{1.008 \times 0.1344}}
  1. Convert the decimals to integers by multiplying the numerator and denominator by appropriate powers of 10.

    0.8064×6.0481.008×0.1344=8064×60481008×1344\sqrt{\frac{0.8064 \times 6.048}{1.008 \times 0.1344}} = \sqrt{\frac{8064 \times 6048}{1008 \times 1344}}
  2. Simplify the terms by division.

    Notice that 8064÷1008=88064 \div 1008 = 8.

    8×1008×60481008×1344=8×60481344\sqrt{\frac{8 \times 1008 \times 6048}{1008 \times 1344}} = \sqrt{\frac{8 \times 6048}{1344}}

    Now, simplify 60481344\frac{6048}{1344}.

    60481344=6048÷13441344÷1344=4.5=92\frac{6048}{1344} = \frac{6048 \div 1344}{1344 \div 1344} = 4.5 = \frac{9}{2}
  3. Substitute the simplified fraction back into the square root.

    8×92\sqrt{8 \times \frac{9}{2}}
  4. Perform the multiplication and take the square root.

    8×92=4×9=36=6\sqrt{\frac{8 \times 9}{2}} = \sqrt{4 \times 9} = \sqrt{36} = 6

    The value is 6\boxed{\text{6}}.

    2. Evaluate;

    −4 of [(−4+−5÷15)+−3−4÷6]84÷−7+3−(−5)\frac{-4 \text{ of } [(-4 + -5 \div 15) + -3 - 4 \div 6]}{84 \div -7 + 3 - (-5)}
  5. Evaluate the numerator.

    First, evaluate the expression inside the brackets: [(−4+−5÷15)+−3−4÷6][(-4 + -5 \div 15) + -3 - 4 \div 6]
    Inside the first parenthesis: (−4+−5÷15)(-4 + -5 \div 15)

    −5÷15=−13-5 \div 15 = -\frac{1}{3} −4+(−13)=−123−13=−133-4 + \left(-\frac{1}{3}\right) = -\frac{12}{3} - \frac{1}{3} = -\frac{13}{3}

    Now substitute this back into the brackets: [−133+−3−4÷6]\left[-\frac{13}{3} + -3 - 4 \div 6\right]
    Perform the division: 4÷6=46=234 \div 6 = \frac{4}{6} = \frac{2}{3}
    The expression in brackets is: $\left[-\frac{13}{3} - 3 - \frac{2

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