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Distribute 2sqrt(3) into the first parenthesis.

Mathematics

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2 parts
a

43−22−24\sqrt{3} - 2\sqrt{2} - 2

b
2.10

Step-by-step solution

7 steps

Part a· 3 steps

Simplify 23(2−3)+22(2−1)2\sqrt{3}(2-\sqrt{3}) + 2\sqrt{2}(\sqrt{2}-1)

  1. Distribute 232\sqrt{3} into the first parenthesis.

    23(2−3)=(23×2)−(23×3)2\sqrt{3}(2-\sqrt{3}) = (2\sqrt{3} \times 2) - (2\sqrt{3} \times \sqrt{3}) =43−2(3×3)= 4\sqrt{3} - 2(\sqrt{3 \times 3}) =43−2(3)= 4\sqrt{3} - 2(3) =43−6= 4\sqrt{3} - 6

  2. Distribute 222\sqrt{2} into the second parenthesis.

    22(2−1)=(22×2)−(22×1)2\sqrt{2}(\sqrt{2}-1) = (2\sqrt{2} \times \sqrt{2}) - (2\sqrt{2} \times 1) =2(2×2)−22= 2(\sqrt{2 \times 2}) - 2\sqrt{2} =2(2)−22= 2(2) - 2\sqrt{2} =4−22= 4 - 2\sqrt{2}

  3. Combine the simplified expressions.

    (43−6)+(4−22)(4\sqrt{3} - 6) + (4 - 2\sqrt{2}) =43−6+4−22= 4\sqrt{3} - 6 + 4 - 2\sqrt{2}

    Combine the constant terms:

    =43−22−2= 4\sqrt{3} - 2\sqrt{2} - 2

    The simplified expression is 43−22−2\boxed{4\sqrt{3} - 2\sqrt{2} - 2}.

Part b2.10

Hence evaluate your answer in (a) correct to 2dp given that 2=1.414\sqrt{2} = 1.414 and 3=1.732\sqrt{3} = 1.732

  1. Substitute the given values of 2\sqrt{2} and 3\sqrt{3} into the simplified expression from part (a).

    43−22−2=4(1.732)−2(1.414)−24\sqrt{3} - 2\sqrt{2} - 2 = 4(1.732) - 2(1.414) - 2

  2. Perform the multiplications.

    4×1.732=6.9284 \times 1.732 = 6.928 2×1.414=2.8282 \times 1.414 = 2.828

  3. Perform the subtractions.

    6.928−2.828−26.928 - 2.828 - 2 =4.100−2= 4.100 - 2 =2.100= 2.100

  4. Round the result to 2 decimal places.

    2.100≈2.102.100 \approx 2.10

    The evaluated answer is 2.10\boxed{2.10}.

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