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Solve the following problems: 1. 3^x = 81 2. sqrt(64) 3. sqrt((16)/(25)) 4. Show 5346 is divisible by 9 without dividing.

Mathematics

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Step-by-step solution

12 steps

3. Solve for x: 3x=813^x = 81

  1. Express 81 as a power of 3.

    81=3×3×3×3=3481 = 3 \times 3 \times 3 \times 3 = 3^4
  2. Substitute this into the equation.

    3x=343^x = 3^4
  3. Since the bases are the same, equate the exponents.

    x=4x = 4

    The value of x is 4\boxed{4}.

    4. Find the square root of 64.

  4. Find a number that, when multiplied by itself, equals 64.

    8×8=648 \times 8 = 64
  5. Therefore, the square root of 64 is 8.

    64=8\sqrt{64} = 8

    The square root of 64 is 8\boxed{8}.

    5. Find the square root of 1625\frac{16}{25}.

  6. Apply the property of square roots for fractions

    ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}.

    1625=1625\sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}}
  7. Find the square root of the numerator (16).

    16=4\sqrt{16} = 4
  8. Find the square root of the denominator (25).

    25=5\sqrt{25} = 5
  9. Combine the results.

    1625=45\frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}

    The square root is 45\boxed{\frac{4}{5}}.

    6. Without dividing, show that 5346 is divisible by 9.

  10. Recall the divisibility rule for 9

    A number is divisible by 9 if the sum of its digits is divisible by 9.

  11. Calculate the sum of the digits of 5346.

    5+3+4+6=185 + 3 + 4 + 6 = 18
  12. Check if the sum of the digits (18) is divisible by 9.

    18÷9=218 \div 9 = 2

    Since 18 is divisible by 9, the number 5346 is also divisible by 9.
    Therefore, 5346 is divisible by 9\boxed{\text{5346 is divisible by 9}}.

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