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Simplify the following expression: -4 of [2 ÷ 84 ÷ 7 - 1]

Mathematics

ScanSolve AI Answer

Answer

586147\frac{586}{147}

45

32\frac{3}{2}

Step-by-step solution

21 steps
  1. Evaluate the expression inside the innermost parentheses.

    (−4+−9+15)=(−13+15)=2(-4 + -9 + 15) = (-13 + 15) = 2
  2. Substitute this value back into the expression.

    −4 of [2÷84÷7−1]-4 \text{ of } [2 \div 84 \div 7 - 1]

    "of" means multiplication. Perform divisions from left to right.

    2÷84=284=1422 \div 84 = \frac{2}{84} = \frac{1}{42} 142÷7=142×7=1294\frac{1}{42} \div 7 = \frac{1}{42 \times 7} = \frac{1}{294}
  3. Substitute this result back into the brackets.

    −4×[1294−1]-4 \times \left[\frac{1}{294} - 1\right] −4×[1294−294294]-4 \times \left[\frac{1}{294} - \frac{294}{294}\right] −4×[−293294]-4 \times \left[-\frac{293}{294}\right]
  4. Perform the final multiplication.

    4×293294=1172294\frac{4 \times 293}{294} = \frac{1172}{294}
  5. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2.

    1172÷2294÷2=586147\frac{1172 \div 2}{294 \div 2} = \frac{586}{147}

    The value of the expression is 586147\boxed{\frac{586}{147}}.


  6. Define the two-digit number using its digits.

    Let the tens digit be tt and the units digit be uu.
    The number can be represented as 10t+u10t + u.

  7. Formulate equations based on the given conditions.

    Condition 1: The product of the digits is 20.

    t×u=20(1)t \times u = 20 \quad (1)

    Condition 2: When the digits are reversed, the new number is greater than the original number by 9.
    The reversed number is 10u+t10u + t.

    10u+t=(10t+u)+910u + t = (10t + u) + 9
  8. Simplify the second equation.

    10u+t=10t+u+910u + t = 10t + u + 9

    Subtract uu and 10t10t from both sides:

    9u−9t=99u - 9t = 9

    Divide by 9:

    u−t=1(2)u - t = 1 \quad (2)

    From (2), we can express uu in terms of tt:

    u=t+1u = t + 1
  9. Substitute the expression for uu into equation (1).

    t×(t+1)=20t \times (t + 1) = 20 t2+t=20t^2 + t = 20

    Rearrange into a quadratic equation:

    t2+t−20=0t^2 + t - 20 = 0
  10. Solve the quadratic equation for tt.

    Factor the quadratic equation:

    (t+5)(t−4)=0(t + 5)(t - 4) = 0

    Possible values for tt are t=−5t = -5 or t=4t = 4.
    Since tt is a digit of a positive two-digit number, it must be a positive integer between 1 and 9.
    Therefore, t=4t = 4.

  11. Find the value of uu using u=t+1u = t + 1.

    u=4+1=5u = 4 + 1 = 5
  12. Form the original two-digit number.

    The number is 10t+u=10(4)+5=4510t + u = 10(4) + 5 = 45.

  13. Verify the conditions.

    Product of digits: 4×5=204 \times 5 = 20 (Correct).
    Reversed number: 54.
    Difference: 54−45=954 - 45 = 9 (Correct).

    The number is 45\boxed{45}.


  14. Rewrite the first term using the exponent rule am+n=am×ana^{m+n} = a^m \times a^n.

    52x+2=52x×525^{2x+2} = 5^{2x} \times 5^2 52x+2=25×52x5^{2x+2} = 25 \times 5^{2x}
  15. Substitute this back into the original equation.

    25×52x−20×52x=62525 \times 5^{2x} - 20 \times 5^{2x} = 625
  16. Factor out the common term 52x5^{2x}.

    (25−20)×52x=625(25 - 20) \times 5^{2x} = 625 5×52x=6255 \times 5^{2x} = 625
  17. Express 625 as a power of 5.

    625=54625 = 5^4
  18. Substitute 545^4 back into the equation.

    5×52x=545 \times 5^{2x} = 5^4
  19. Use the exponent rule am×an=am+na^m \times a^n = a^{m+n} on the left side.

    51+2x=545^{1+2x} = 5^4
  20. Since the bases are equal, equate the exponents.

    1+2x=41 + 2x = 4
  21. Solve for xx.

    2x=4−12x = 4 - 1 2x=32x = 3 x=32x = \frac{3}{2}

    The solution for xx is 32\boxed{\frac{3}{2}}.

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