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Evaluate (2a2)/(c-b) given a=-3, b=2, c=5. Subtract 14346 from 22436 and give the answer in base ten.

Mathematics

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: Evaluate 2a2c−b\frac{2a^2}{c-b}

Given a=−3a = -3, b=2b = 2, and c=5c = 5.

  1. Substitute the given values into the expression.

    2a2c−b=2(−3)25−2\frac{2a^2}{c-b} = \frac{2(-3)^2}{5-2}
  2. Calculate the numerator.

    2(−3)2=2(9)=182(-3)^2 = 2(9) = 18
  3. Calculate the denominator.

    5−2=35-2 = 3
  4. Divide the numerator by the denominator.

    183=6\frac{18}{3} = 6

    The evaluated expression is 6\boxed{\text{6}}.

    : Subtract 143461434_6 from 224362243_6 and give the answer in base ten.

  5. Subtract 143461434_6 from 224362243_6 in base 6.

    We perform column subtraction:
    224362243_6

    • 143461434_6

    Starting from the rightmost column (units place):
    3−43 - 4: We cannot subtract 4 from 3. Borrow from the next column (the 4 in 224362243_6).
    The 4 becomes 3, and we add 6 (the base) to the 3, making it 3+6=93+6=9.
    Now, 9−4=59 - 4 = 5. So the units digit is 5.

    Next column:
    3−3=03 - 3 = 0. So the next digit is 0.

    Next column:
    2−42 - 4: We cannot subtract 4 from 2. Borrow from the next column (the 2 in 224362243_6).
    The 2 becomes 1, and we add 6 to the 2, making it 2+6=82+6=8.
    Now, 8−4=48 - 4 = 4. So the next digit is 4.

    Next column:
    1−1=01 - 1 = 0. So the leftmost digit is 0.

    The result of the subtraction in base 6 is 040560405_6, which is 4056405_6.

  6. Convert the result 4056405_6 to base ten.

    To convert a number from base bb to base 10, we use the formula:

    (dndn−1…d1d0)b=dn×bn+dn−1×bn−1+⋯+d1×b1+d0×b0(d_n d_{n-1} \dots d_1 d_0)_b = d_n \times b^n + d_{n-1} \times b^{n-1} + \dots + d_1 \times b^1 + d_0 \times b^0

    For 4056405_6:

    4056=4×62+0×61+5×60405_6 = 4 \times 6^2 + 0 \times 6^1 + 5 \times 6^0 =4×36+0×6+5×1= 4 \times 36 + 0 \times 6 + 5 \times 1 =144+0+5= 144 + 0 + 5 =149= 149

    The answer in base ten is 149\boxed{\text{149}}.

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