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Algebra

Workshop Calculation and Science, Module-3

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By Roni
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Algebra


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Q1: What is the value of \(14x+3y+25x+2y\)?

  • \(17x+27y\)
  • \(16x+28y\)
  • \(39x+5y\)
  • \(44xy\)
Show Calculation \[ 14x+3y+25x+2y \]\[ = (14x+25x) + (3y+2y) \] \[ = 39x + 5y \]

Q2: What is the multiplication value of \(5a^2b \times 8a^5b^3\)?

  • \(40a^{7}b^{4}\)
  • \(40a^{7}b^{2}\)
  • \(40a^{2}b^{4}\)
  • \(40a^{10}b^{3}\)
Show Calculation \[ 5a^2b \times 8a^5b^3 = (5\times8)\,a^{2+5}\,b^{1+3} = 40a^7b^4 \]

Q3: What is the simplified value of \(\dfrac{3x + 15}{5x + 25}\)?

  • \(\tfrac{5}{3}\)
  • \(\tfrac{3}{5}\)
  • -\(\tfrac{5}{3}\)
  • -\(\tfrac{3}{5}\)
Show Calculation \[ \frac{3x+15}{5x+25} = \frac{3(x+5)}{5(x+5)} = \frac{3}{5} \]

Q4: What is the value of x if \(13+x=20\)?

  • 8
  • 7
  • 9
  • 13
Show Calculation \[ 13 + x = 20 \Rightarrow x = 20 - 13 = 7 \]

Q5: What is the value of x, if \(x(120) = 960\)?

  • 6
  • 7
  • 8
  • 10
Show Calculation \[ 120x = 960 \Rightarrow x = \frac{960}{120} = 8 \]

Q6: What is the formula for \(a^m \times a^n\)?

  • \(a^{m+n}\)
  • \(a^{m-n}\)
  • \(a^{mn}\)
  • \(n a^m\)

Q7: Which is the formula for \(\dfrac{a^m}{a^n}\)?

  • \(a^{m+n}\)
  • \(a^{m-n}\)
  • \(a^{mn}\)
  • \((a)^{m/n}\)

Q8: What is the value of any number raised to the power of 0?

  • 0
  • 1
  • α
  • 10

Q9: What is the value of \(\dfrac{1}{a^m}\)?

  • \(a^m\)
  • \(a^{-m}\)
  • \(\sqrt[m]{a}\)
  • \(\sqrt[m]{a}\)

Q10: Which is equal to \((a^m)^n\)?

  • \(a^{m+n}\)
  • \(a^m\)
  • \(a^{m-n}\)
  • \(a^{mn}\)

Q11: What is the expanded form of \((a+b)^2\)?

  • \(a^2 + 2ab + b^2\)
  • \(a^2 - 2ab + b^2\)
  • \(a^2 + 2ab - b^2\)
  • -\(a^2 + 2ab - b^2\)

Q12: What is the formula for \((a-b)^2\)?

  • \(a^2 - 2ab + b^2\)
  • \(a^2 + 2ab + b^2\)
  • \(a^2 - 2ab - b^2\)
  • -\(a^2 - 2ab - b^2\)

Q13: Which is equal to \((a+b)^2 - (a-b)^2\)?

  • \(2ab\)
  • \(3ab\)
  • \(4ab\)
  • \(5ab\)
Show Calculation \[ (a+b)^2 - (a-b)^2\]\[ = \big(a^2+2ab+b^2\big) - \big(a^2-2ab+b^2\big) = 4ab \]

Q14: What is the value of \(a^5 \times a^{-3} \times a^{-2}\)?

  • \(a^{10}\)
  • \(a^{-10}\)
  • \(a^{6}\)
  • \(a^{0}\)
Show Calculation \[ a^5 \times a^{-3} \times a^{-2} = a^{5-3-2} = a^0 = 1 \]

Q15: What is the value of \((a^3)^5\)?

  • \(a^{15}\)
  • \(a^{8}\)
  • \(a^{2}\)
  • \(a^{-2}\)
Show Calculation \[ (a^3)^5 = a^{3\times5} = a^{15} \]

Q16: What is the value of \(625^{1/2}\)?

  • 0
  • 25
  • 525
  • 62.5
Show Calculation \[ 625^{1/2} = \sqrt{625} = 25 \]

Q17: What is the value of \(\dfrac{1}{a^{-5}}\)?

  • \(a^{5}\)
  • \(1/a^{5}\)
  • \(5a\)
  • -\(5a\)

Q18: What is the value of \(\dfrac{5x^4}{5x^3}\)?

  • \(5x\)
  • \(5x^{7}\)
  • \(x\)
  • \(5x^{4/3}\)
Show Calculation \[ \frac{5x^4}{5x^3} = \frac{5}{5}\,x^{4-3} = x \]

Q19: What is the subtracted value of \(3x - 4x^2 + 2y^2\) from \(4y^2 - 2x + 8x^2\)?

  • \(2y^2 - 5x + 12x^2\)
  • \(2y^2 + 5x - 12x^2\)
  • \(2y^2 - 5x - 12x^2\)
  • -\(2y^2 - 5x + 12x^2\)
Show Calculation \[ (4y^2 - 2x + 8x^2) - (3x - 4x^2 + 2y^2) \] \[ = 4y^2 - 2x + 8x^2 - 3x + 4x^2 - 2y^2 \] \[ = (4y^2-2y^2) + (-2x-3x) + (8x^2+4x^2)\]\[ = 2y^2 - 5x + 12x^2 \]

Q20: What is the value of adding \((5x+2y)\), \((4x-7z)\) and \((15z - 3y)\)?

  • \(9x - y + 8z\)
  • \(x - 9y + 8z\)
  • \(x + 9y + 8z\)
  • \(9x + y - 8z\)
Show Calculation \[ (5x+2y) + (4x-7z) + (15z-3y)\]\[ = (5x+4x) + (2y-3y) + (-7z+15z) \] \[ = 9x - y + 8z \]

Q21: What is the value of \(\dfrac{12x^3y^2}{4x^2y}\)?

  • \(8xy\)
  • \(16xy\)
  • \(3xy\)
  • -3xy
Show Calculation \[ \frac{12x^3y^2}{4x^2y} = \frac{12}{4} \, x^{3-2} \, y^{2-1} = 3xy \]

Q22: What is the value of x, if \(3(2x-4) = -4x + 28\)?

  • 4
  • 8
  • 6
  • 12
Show Calculation \[ 3(2x-4) = -4x + 28 \]\[\Rightarrow 6x - 12 = -4x + 28 \] \[ 6x + 4x = 28 + 12 \Rightarrow 10x = 40 \Rightarrow x = 4 \]

Q23: What is the value of x if \(\dfrac{x+2}{2} = 19\)?

  • 38
  • 33
  • 35
  • 36
Show Calculation \[ \frac{x+2}{2}=19 \Rightarrow x+2 = 38 \Rightarrow x = 36 \]

Q24: What is the value of x if \(11x + 4 = 37\)?

  • 2
  • 3
  • 4
  • 5
Show Calculation \[ 11x + 4 = 37 \Rightarrow 11x = 33 \Rightarrow x = 3 \]

Q25: What is the value of \(\dfrac{1}{a^{-m}}\)?

  • \(a^{-m}\)
  • \(a^{m}\)
  • \(\sqrt[m]{a}\)
  • \(a^{1/m}\)

Q26: What is the value of \(\dfrac{a^m}{a^n}\)?

  • \(a^{m+n}\)
  • \(\sqrt{a^{m+n}}\)
  • \(\sqrt{a^{m-n}}\)
  • \(a^{m-n}\)

Q27: Which is the expansion of \(a^3 + b^3\)?

  • \((a-b)(a^2 + b^2 - ab)\)
  • \((a+b)(a^2 - ab + b^2)\)
  • \(a^3 + b^3 + 3ab(a+b)\)
  • \(a^3 - b^3 + 3ab(a-b)\)

Q28: What is the expansion of \((a+b+c)^2\)?

  • \(a^2+b^2+c^2+2(ab+bc+ca)\)
  • \(a^2+b^2+c^2-2ab+2bc+2ca\)
  • \(a^2+b^2+c^2+2ab-2bc+2ca\)
  • \(a^2-b^2-c^2+2ab+2bc+2ca\)

Q29: Which is expanded form of \(a^3 - b^3\)?

  • \((a+b)(a^2 - b^2 - ab)\)
  • \((a-b)(a^2 + ab + b^2)\)
  • \((a-b)(a^2 - b^2 - ab)\)
  • \((a-b)(a^2 - b^2 + ab)\)

Q30: What is the value of \(\dfrac{6^3}{(-3)^3}\)?

  • 8
  • -8
  • 27
  • -27
Show Calculation \[ \frac{6^3}{(-3)^3} \] \[ = \frac{6 \times 6 \times 6}{(-3) \times (-3) \times (-3)} \] \[ = \frac{216}{-27} \] \[ = -8 \]

Q31: What is the value of \(x^2 - y^2\) if \((x+y) = 9\), \((x - y) = 4\)?

  • 13
  • 65
  • 36
  • 46
Show Calculation \[ x^2 - y^2 = (x+y)(x-y) = 9 \times 4 = 36 \]

Q32: What is the value of \(x\) if \(x - y = 6\) and \(x + y = 8\)?

  • 5
  • 6
  • 7
  • 14
Show Calculation \[ (x-y)+(x+y) = 6 + 8 \]\[\Rightarrow 2x = 14 \Rightarrow x = 7 \]

Q33: What is the value of \(a^2+b^2\) if \(a+b=9\) and \(ab = 20\)?

  • 121
  • -121
  • 41
  • -41
Show Calculation \[ a^2 + b^2 = (a+b)^2 - 2ab \]\[= 9^2 - 2\cdot20 = 81 - 40 = 41 \]

Q34: What is the value of \(ab\) if \((a+b)^2=36\) and \((a-b)^2=24\)?

  • 6
  • 4
  • 3
  • 2
Show Calculation \[ (a+b)^2 - (a-b)^2 = 4ab \]\[\Rightarrow 36 - 24 = 12 = 4ab \] \[ \Rightarrow ab = \frac{12}{4} = 3 \]

Q35: What is the value of \(x^3+3y^2x^2\) if \(x=3, y=2\)?

  • 135
  • 81
  • 54
  • 63
Show Calculation \[ x^3 + 3y^3x = x^3 + 3(x^2)(y^2) \] Substitute \(x=3,\ y=2\): \[ x^3 = 3^3 = 27, y^2 = 2^2 = 4, x^2 = 9 \] \[ 3 \cdot x^2 \cdot y^2 = 3 \cdot 9 \cdot 4 = 108 \] \[ \Rightarrow x^3 + 3y^2x^2 = 27 + 108 = 135 \]

Q36: What are the three consecutive numbers if their sum is 42?

  • 11, 12, 13
  • 12, 13, 14
  • 13, 14, 15
  • 14, 15, 16
Show Calculation Let the numbers be \(n, n+1, n+2\). Then \[ n + (n+1) + (n+2) = 3n + 3 = 42 \]\[\Rightarrow 3n = 39 \Rightarrow n = 13 \] So the numbers are \(13, 14, 15\).
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