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In the given grid, find the missing number: [32, 8; 128, ?]

Mathematics

ScanSolve AI Answer

Step-by-step solution

6 steps
  1. Express the given numbers as powers of a common base.

    The numbers in the grid are 32, 8, and 128. These can all be expressed as powers of 2.

    32=258=23128=2732 = 2^5 \\ 8 = 2^3 \\ 128 = 2^7

    Let the missing number be XX. We assume XX is also a power of 2, so X=2xX = 2^x.

  2. Form a grid of the exponents.

    The grid of numbers is:

    [328128?]\begin{bmatrix} 32 & 8 \\ 128 & ? \end{bmatrix}

    Replacing the numbers with their exponents (base 2):

    [537x]\begin{bmatrix} 5 & 3 \\ 7 & x \end{bmatrix}
  3. Identify the pattern in the exponents.

    Observe the sum of the exponents in the first row:

    5+3=85 + 3 = 8

    Assume the same pattern applies to the second row, meaning the sum of the exponents in the second row is also 8.

    7+x=87 + x = 8
  4. Solve for the missing exponent xx.

    x=8−7x=1x = 8 - 7 \\ x = 1
  5. Calculate the missing number XX.

    Since X=2xX = 2^x and x=1x = 1:

    X=21X=2X = 2^1 \\ X = 2
  6. Compare the result with the given options.

    The calculated missing number is 2.
    The options are:
    A. 4
    B. 3
    C. 6
    D. 2
    E. 8

    The calculated value matches option D.

    The final answer is D. 2\boxed{\text{D. 2}}.

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