surds-and-indices.json

{
  "formula": "SURD:\n\nThe irrational root of an integer is called a surd. Eg. √2, √3, √5 etc..\n\nINDEX:\n\nThe power or exponent is called an index. Eg. in 3², 3 is the base and 2 is the index.\n\nLAWS OF INDICES:\n\n(i)a^m * a^n = a^(m+n)\n(ii)a^m ÷ a^n = a^(m-n)\n(iii)(a^m)^n = a^(mn)\n(iv)(ab)^n = a^n * b^n\n(v)(a/b)^n = (a^n)/(b^n)\n(vi)a^0 = 1\n(vii)a^1 = a",
  "problems": [
    {
      "id": 1,
      "answerIndex": 1,
      "answers": ["2", "4", "6", "12"],
      "question": "If √2^n = 4, then the value of n is",
      "solution": "√2^n = 4\n\n2^(n/2) = 2^2\n\nn/2 = 2\n\nn = 4"
    },
    {
      "id": 2,
      "answerIndex": 3,
      "answers": ["25", "125", "625", "5"],
      "question": "If 5^y= 3125, then the value of 5^(y-4) is",
      "solution": "5^5 = 3125\n\nSo, y = 5\n\n5^(y-4) = 5^1 = 5"
    },
    {
      "id": 3,
      "answerIndex": 2,
      "answers": ["2.29", "2.75", "4.75", "4.5"],
      "question": "(20)^3.25 x (20)^x = 20^8. Find x.",
      "solution": "(20)^3.25 x (20)^x = 20^8\n\n(20)^(3.25+x) = 20^8\n\n3.25+x = 8\n\nx = 4.75"
    },
    {
      "id": 4,
      "answerIndex": 2,
      "answers": ["1.45", "1.88", "2.9", "3.7"],
      "question": "Given that 10^0.48 = x, 10^0.70 = y and x^z = y^2, then the value of z is close to",
      "solution": "10^0.48 = x\n10^0.70 = y\n\nx^z = y^2\n\n10^(0.48z) = 10^(0.7*2)\n\n0.48z = 1.4\n\nz = 1.4/0.48 = 2.9167"
    },
    {
      "id": 5,
      "answerIndex": 2,
      "answers": ["0", "2", "4", "6"],
      "question": "If 3^(x - y) = 27 and 3^(x + y) = 243, then x is equal to",
      "solution": "3^(x - y) = 27\nx-y = 3 ---- (I)\n\n3^(x + y) = 243\nx+y = 5 ----(II)\n\nSolving (I) and (II),\n2x = 8\nx = 4"
    },
    {
      "id": 6,
      "answerIndex": 0,
      "answers": ["3", "9", "27", "81.25"],
      "question": "(81)^0.16 x (81)^0.09 = ?",
      "solution": "(81)^0.16 x (81)^0.09\n= (81)^(0.16+0.09)\n= 81^0.25\n= (3^4)^0.25\n= 3^(4*0.25)\n= 3"
    },
    {
      "id": 7,
      "answerIndex": 3,
      "answers": ["25", "125", "64", "8"],
      "question": "(0.25)^-1.5 = ?",
      "solution": "(0.25)^-1.5\n= (1/4)^-1.5\n= 4^1.5\n= (2^2)^1.5\n= 2^3\n= 8"
    },
    {
      "id": 8,
      "answerIndex": 2,
      "answers": ["7/16 \n", "7 / 32", "7/48", "7/64"],
      "question": "If 2^x = 4^y =8^z and (1/2x +1/4y +1/6z) =24/7 then the value of 'z' is",
      "solution": "2^x = 4^y =8^z\n\n2^x = 2^(2y) = 2^(3z)\n\nx = 2y = 3z\n\nx = 3z\ny = 3z/2\n\n(1/2x +1/4y +1/6z) =24/7\n\n1/6z + 1/6z + 1/6z = 24/7\n\n1/2z = 24/7\n\nz = 7/48"
    },
    {
      "id": 9,
      "answerIndex": 1,
      "answers": ["343", "216", "64", "243"],
      "question": "If a and b are whole numbers such that a^b = 49, the value of (a - 1)^(b + 1) is",
      "solution": "a^b = 49\n7^2 = 49\na = 7, b = 2\n\n(a-1)^(b+1)\n= 6^3\n= 216"
    },
    {
      "id": 10,
      "answerIndex": 0,
      "answers": ["1", "0", "xyz", "√xyz"],
      "question": "If x= ya , y = zb  , z = xc  then find the value\nof abc.",
      "solution": "a = x/y\nb = y/z\nc = z/x\n\nabc = (x/y)*(y/z)*(z/x) = 1\n"
    },
    {
      "id": 11,
      "answerIndex": 3,
      "answers": ["1000", "10000", "10^5", "10^6"],
      "question": "x = 10^175 : 10^169. Find x.",
      "solution": "x = 10^175 : 10^169\n   = 10^(175-169)\n   = 10^6"
    },
    {
      "id": 12,
      "answerIndex": 2,
      "answers": ["125", "625", "3125", "250"],
      "question": "(0.04)^-2.5 = ?",
      "solution": "(0.04)^(-2.5)\n= (4/100)^(-5/2)\n= (1/25)^(-5/2)\n= 25^(5/2)\n= 5^(2*5/2)\n= 5^5\n= 3125"
    },
    {
      "id": 13,
      "answerIndex": 1,
      "answers": ["2", "4", "5", "6"],
      "question": "(243)^(x/5) = 81. Find x.",
      "solution": "(243)^(x/5) = 81\n\n3^5 = 243\n\n3^x = 81\n\nx = 4"
    },
    {
      "id": 14,
      "answerIndex": -1,
      "answers": ["6", "7", "8", "10"],
      "question": "2^a * 2^b = 128. Find the sum of a and b.",
      "solution": "2^a * 2^b = 128\n\n2^(a+b) = 2^7\n\na+b = 7"
    },
    {
      "id": 15,
      "answerIndex": 0,
      "answers": ["7", "1", "6", "3√2"],
      "question": "If x = 3+√2, y= 3-√2, find the product of x and y.",
      "solution": "xy\n= (3+√2)*(3-√2)\n= 9-2\n= 7"
    },
    {
      "id": 16,
      "answerIndex": 3,
      "answers": ["1:3", "2:1", "3:1", "4:1"],
      "question": "If √3^p = 9^q, find the ratio of p and q.",
      "solution": "√3^p = 9^q\n\n3^(p/2) = 3^2q\n\np/2 = 2q\n\np:q = 4:1"
    },
    {
      "id": 17,
      "answerIndex": 0,
      "answers": ["10.45", "7.85", "5.15", "4.25"],
      "question": "(15)^m : (15)^2.65 = (15)^7.8. Find m.",
      "solution": "(15)^m : (15)^2.65 = (15)^7.8\n\nm-2.65 = 7.8\n\nm = 10.45"
    },
    {
      "id": 18,
      "answerIndex": 0,
      "answers": ["1", "3", "0.3", "0.67"],
      "question": "Given m = 12^0.27, n = 12^0.81 and n^p=m³. Find the value of p.",
      "solution": "m = 12^0.27, n = 12^0.81, n^p=m³\n\nm³ = 12^(0.27*3) = 12^0.81\n\np = 1"
    },
    {
      "id": 19,
      "answerIndex": 0,
      "answers": ["8", "4", "2", "3.2"],
      "question": "(32)^1.1 - (32)^0.9 = ?",
      "solution": "(32)^1.5 : (32)^0.9\n= (32)^(1.5-0.9)\n= (32)^0.6\n= (2^5)^0.6\n= 2^3\n= 8"
    },
    {
      "id": 20,
      "answerIndex": 2,
      "answers": ["9/5", "2/9", "8/5", "4/9"],
      "question": "If 3^a = 27^b and (1/3a +1/9b) = 5/36, find the value of b.",
      "solution": "3^a = 3^(3b)\n\na = 3b\n\n(1/3a)+(1/9b) = 5/36\n(1/9b)+(1/9b) = 5/36\n\n2/9b = 5/36\n\nb = 8/5"
    }
  ]
}