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Evaluate sqrt((0.0018 * 0.12) / (0.05 * 0.2)). Leaving your answer in the form K x 10ⁿ. Hence or otherwise find the value of (k+n)³

Mathematics

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Answer

1.47×10−1\text{1.47} \times 10^{-1}

0.104\text{0.104}

Step-by-step solution

7 steps

  1. Convert all numbers in the expression to scientific notation.

    0.0018=1.8×10−30.0018 = 1.8 \times 10^{-3} 0.12=1.2×10−10.12 = 1.2 \times 10^{-1} 0.05=5×10−20.05 = 5 \times 10^{-2} 0.2=2×10−10.2 = 2 \times 10^{-1}

  2. Substitute these scientific notations into the given expression.

    (1.8×10−3)×(1.2×10−1)(5×10−2)×(2×10−1)\sqrt{\frac{(1.8 \times 10^{-3}) \times (1.2 \times 10^{-1})}{(5 \times 10^{-2}) \times (2 \times 10^{-1})}}

  3. Simplify the numerator and the denominator separately.

    Numerator: Multiply the numerical parts and the powers of 10.

    (1.8×1.2)×(10−3×10−1)=2.16×10−3−1=2.16×10−4(1.8 \times 1.2) \times (10^{-3} \times 10^{-1}) = 2.16 \times 10^{-3-1} = 2.16 \times 10^{-4}

    Denominator: Multiply the numerical parts and the powers of 10.

    (5×2)×(10−2×10−1)=10×10−2−1=10×10−3=1×10−2(5 \times 2) \times (10^{-2} \times 10^{-1}) = 10 \times 10^{-2-1} = 10 \times 10^{-3} = 1 \times 10^{-2}

  4. Substitute the simplified numerator and denominator back into the expression and simplify the fraction.

    2.16×10−41×10−2=2.16×10−4−(−2)=2.16×10−2\sqrt{\frac{2.16 \times 10^{-4}}{1 \times 10^{-2}}} = \sqrt{2.16 \times 10^{-4 - (-2)}} = \sqrt{2.16 \times 10^{-2}}

  5. Take the square root of the simplified expression.

    2.16×10−2=2.16×10−2=2.16×10−1\sqrt{2.16 \times 10^{-2}} = \sqrt{2.16} \times \sqrt{10^{-2}} = \sqrt{2.16} \times 10^{-1}

  6. Evaluate 2.16\sqrt{2.16} and express the answer in the form K×10nK \times 10^n.

    2.16≈1.4696938...\sqrt{2.16} \approx 1.4696938...

    So, the expression is approximately 1.4696938×10−11.4696938 \times 10^{-1}.
    To express this in the form K×10nK \times 10^n where 1≤K<101 \le K < 10, we round KK to a reasonable number of significant figures, for example, three significant figures.

    K≈1.47K \approx 1.47 n=−1n = -1

    Thus, the answer in the form K×10nK \times 10^n is 1.47×10−1\boxed{\text{1.47} \times 10^{-1}}.

  7. Find the value of (K+n)3(K+n)^3.

    Substitute the values of KK and nn:

    K+n=1.47+(−1)=0.47K+n = 1.47 + (-1) = 0.47

    Now, calculate (K+n)3(K+n)^3:

    (K+n)3=(0.47)3=0.103823(K+n)^3 = (0.47)^3 = 0.103823

    Rounding to three significant figures, we get:

    (K+n)3≈0.104(K+n)^3 \approx \boxed{\text{0.104}}
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