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Evaluate (14702)/(sqrt(7056)) using prime factorization. What are all prime numbers less than ten?

Mathematics

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Step-by-step solution

7 steps

  1. Express 1470 as a product of its prime factors.

    1470=2×7351470 = 2 \times 735 735=3×245735 = 3 \times 245 245=5×49245 = 5 \times 49 49=7×749 = 7 \times 7

    So, the prime factorization of 1470 is:

    1470=2×3×5×721470 = 2 \times 3 \times 5 \times 7^2

  2. Express 7056 as a product of its prime factors.

    7056=2×35287056 = 2 \times 3528 3528=2×17643528 = 2 \times 1764 1764=2×8821764 = 2 \times 882 882=2×441882 = 2 \times 441 441=3×147441 = 3 \times 147 147=3×49147 = 3 \times 49 49=7×749 = 7 \times 7

    So, the prime factorization of 7056 is:

    7056=24×32×727056 = 2^4 \times 3^2 \times 7^2

  3. Evaluate 147027056\frac{1470^2}{\sqrt{7056}} in prime factor form.

    First, find 147021470^2:

    14702=(2×3×5×72)2=22×32×52×741470^2 = (2 \times 3 \times 5 \times 7^2)^2 = 2^2 \times 3^2 \times 5^2 \times 7^4

    Next, find 7056\sqrt{7056}:

    7056=24×32×72=242×322×722=22×31×71\sqrt{7056} = \sqrt{2^4 \times 3^2 \times 7^2} = 2^{\frac{4}{2}} \times 3^{\frac{2}{2}} \times 7^{\frac{2}{2}} = 2^2 \times 3^1 \times 7^1

    Now, substitute these into the expression:

    147027056=22×32×52×7422×31×71\frac{1470^2}{\sqrt{7056}} = \frac{2^2 \times 3^2 \times 5^2 \times 7^4}{2^2 \times 3^1 \times 7^1}

    Simplify by subtracting the exponents of common prime factors:

    =22−2×32−1×52×74−1= 2^{2-2} \times 3^{2-1} \times 5^2 \times 7^{4-1} =20×31×52×73= 2^0 \times 3^1 \times 5^2 \times 7^3 =1×3×52×73= 1 \times 3 \times 5^2 \times 7^3 3×52×73\boxed{3 \times 5^2 \times 7^3}

    2) a)

  4. Identify all prime numbers less than ten.

    The prime numbers less than ten are 2, 3, 5, and 7.

  5. Arrange these prime numbers in descending order to form a number.

    Descending order means from largest to smallest: 7, 5, 3, 2.
    The number formed is 7532.

    7532\boxed{7532}

    2) b)

  6. Identify the second digit of the number formed.

    The number formed is 7532. The digits are:

    • First digit: 7
    • Second digit: 5
    • Third digit: 3
    • Fourth digit: 2

  7. Determine the total value of the second digit.

    The second digit is 5, and it is in the hundreds place.
    Total value = 5×100=5005 \times 100 = 500.

    500\boxed{500}
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