Maths

• What is $2/5 *$$1/5 +$$3/4$ expressed with a common denominator of 20?


  $4/20 +$$15/20 =$$19/20$
• How to evaluate $2/3 *$$19/20$ and simplify the result?


  1. Multiply numerators: $2 *$$19 =$$38$
  2. Multiply denominators: $3 *$$20 =$$60$
  3. Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2:

  $38/60 =$$19/30$
• How to divide fractions: $2\frac{1}{6} \div \frac{8}{9}?$


  1. Convert mixed number to improper fraction: $2\frac{1}{6} =$$\frac{13}{6}$
  2. Multiply by the reciprocal of the divisor: $\frac{13}{6} \times \frac{9}{8}$
  3. Multiply numerators and denominators: $13 \times 9 =$$117,$ $6 \times 8 =$$48$
  4. Simplify the fraction: $\frac{117}{48} =$$\frac{39}{16}$
  5. Convert improper fraction to mixed number: $\frac{39}{16} =$$2\frac{7}{16}$
• What is $13 \times 9?$


  $117$
• What is $6 \times 8?$


  $48$
• How to simplify the fraction $117/48?$


  • Divide both numerator and denominator by 3:

  $117 \div 3 =$$39$

  $48 \div 3 =$$16$

  • Final simplified fraction: $39/16$
• What is $39/16$ as a mixed number?


  $2\frac{7}{16}$
• What is the square root of 20 simplified?

  $2\sqrt{5}$
• How can the expression $10 - 2\sqrt{5} +$$8\sqrt{5}$ be simplified?


  $10 +$$6\sqrt{5}$
• How do you rationalize the denominator of $12/\sqrt{15}?$


  1. Multiply the fraction by $\frac{\sqrt{15}}{\sqrt{15}}:$

  $\frac{12}{\sqrt{15}} \times \frac{\sqrt{15}}{\sqrt{15}} =$$\frac{12\sqrt{15}}{15}$

  1. Simplify the fraction by dividing numerator and denominator by 3:

  $\frac{12\sqrt{15}}{15} =$$\frac{4\sqrt{15}}{5}$
• What is $15 \div 3?$


  $5$
• What is $12\sqrt{15} \div 3?$


  $4\sqrt{15}$
• How do you simplify the expression $5c^2 \div c^3 \times c^4?$


  1. Multiply terms with the same base by adding the powers: $c^3 \times c^4 =$$c^{3+4} =$$c^7$
  2. Divide terms with the same base by subtracting the powers: $c^2 \div c^7 =$$c^{2-7} =$$c^{-9}$
  3. Final simplified expression: $5c^{-9}$
  4. Express as a positive power: $5/c^9$
• What is $8^{5/3}$ evaluated?


  $32$
• What does $c^{-9}$ equal in positive power notation?


  $1/c^9$
• What are the steps to evaluate $8^{5/3}?$


  1. Rewrite the exponent as a root: $( \sqrt[3]{8} )^5$
  2. Calculate the cube root of 8: $\sqrt[3]{8} =$$2$
  3. Raise 2 to the power of 5: $2^5 =$$32$
• How to simplify the expression $\frac{a^4 \times 3a}{\sqrt{a}}?$


  1. Multiply terms on the numerator: $a^4 \times 3a =$$3a^5$
  2. Divide by $\sqrt{a},$ which is $a^{1/2}:$ $3a^5 \div a^{1/2} =$$3a^{5 - 1/2} =$$3a^{9/2}$
  3. Final simplified expression: $3a^{9/2}$
• What is $5 - 1/2?$


  $4\frac{1}{2} \text{ or } 9/2$
• How do you express the expression $3a^{9/2}$ as a product of integers and roots?


  • $a^{9/2}$ can be written as $(a^{1/2})^9 =$$(\sqrt{a})^9$
  • Final expression: $3(\sqrt{a})^9$
• How to expand and simplify $(3x - 2)(2x^2 + 5x - 1)?$


  • Multiply each term in the first bracket by each term in the second bracket:

  $3x(2x^2) =$$6x^3$

  $3x(5x) =$$15x^2$

  $3x(-1) =$$-3x$

  $-2(2x^2) =$$-4x^2$

  $-2(5x) =$$-10x$

  $-2(-1) =$$2$

  • Combine like terms:

  $6x^3 +$$11x^2 - 13x +$$2$
• How can the expression $(x + 7)^2$ be expanded and simplified?


  • Use the perfect square trinomial formula: $(a + b)^2 =$$a^2 +$$2ab +$$b^2$
  • Here, $a =$$x$ and $b =$$7:$

  $(x + 7)^2 =$$x^2 +$$2(x)(7) +$$7^2 =$$x^2 +$$14x +$$49$
• How do you factorize the expression $3x^2 - 48?$


  1. Find the greatest common factor (GCF) of the terms, which is 3:

  $3(x^2 - 16)$

  1. Recognize the difference of two squares in the parentheses:

  $x^2 - 16 =$$(x + 4)(x - 4)$

  1. Final factored expression: $3(x + 4)(x - 4)$
• What are the factors of 24 that add to -11?

  • 11 and 24 can be factored as:
  • (-3) and (-8): $-3 +$$(-8) =$$-11,$ $-3 \times -8 =$$24$
• How to solve the quadratic equation $x^2 - 11x +$$24 =$$0$ by factorizing?


  1. Factor the quadratic expression: $(x - 3)(x - 8) =$$0$
  2. Set each factor equal to 0:

  $x - 3 =$$0 \implies x =$$3$

  $x - 8 =$$0 \implies x =$$8$

  1. Solutions: $x =$$3,$ $x =$$8$
• How to express $7x - 3 - 2x$ as a fraction in its simplest form?


  1. Combine like terms: $7x - 2x - 3 =$$5x - 3$
  2. Divide the expression by $x$ to form a fraction: $\frac{5x - 3}{x}$
• How do you subtract algebraic fractions: $\frac{4x - 2}{x + 2} - \frac{3x + 5}{x - 2}?$


  1. Find the common denominator: $(x + 2)(x - 2)$
  2. Rewrite the fractions with the common denominator:

  $\frac{(4x - 2)(x - 2)}{(x + 2)(x - 2)} - \frac{(3x + 5)(x + 2)}{(x - 2)(x + 2)}$

  1. Expand the numerators:

  $\frac{4x^2 - 8x - 2x + 4 - (3x^2 + 6x + 5x + 10)}{(x + 2)(x - 2)} =$$\frac{4x^2 - 10x + 4 - 3x^2 - 11x - 10}{x^2 - 4}$

  1. Combine like terms in the numerator:

  $\frac{x^2 - 21x - 6}{x^2 - 4}$
• How to simplify the algebraic fraction $\frac{4x - 2}{x + 2} \div \frac{3x + 5}{x - 2}?$


  1. Multiply by the reciprocal of the divisor: $\frac{4x - 2}{x + 2} \times \frac{x - 2}{3x + 5}$
  2. Multiply numerators and denominators: $(4x - 2)(x - 2) =$$4x^2 - 8x - 2x +$$4 =$$4x^2 - 10x +$$4,$ $(x + 2)(3x + 5) =$$3x^2 +$$5x +$$6x +$$10 =$$3x^2 +$$11x +$$10$
  3. Final simplified fraction: $\frac{4x^2 - 10x + 4}{3x^2 + 11x + 10}$
• How to simplify the algebraic fraction $\frac{4x - 2}{x + 2} \div \frac{5}{x - 2}?$


  1. Multiply by the reciprocal of the divisor: $\frac{4x - 2}{x + 2} \times \frac{x - 2}{5}$
  2. Multiply numerators and denominators: $(4x - 2)(x - 2) =$$4x^2 - 10x +$$4,$ $(x + 2)(5) =$$5x +$$10$
  3. Final simplified fraction: $\frac{4x^2 - 10x + 4}{5x + 10}$
• How to factorize the quadratic expression $x^2 +$$x - 20?$


  • Look for two numbers that multiply to -20 and add to 1:
  • Numbers: 5 and -4
  • Factored expression: $(x + 5)(x - 4)$
• How to simplify the algebraic fraction $\frac{x^2 - 16}{2x^2 + x - 10}?$


  1. Factorize the numerator: $x^2 - 16 =$$(x + 4)(x - 4)$
  2. Factorize the denominator:

  • Product: $2(-10) =$$-20$
  • Numbers: -4 and 5
  • Factored expression: $2x^2 +$$5x - 4x - 10 =$$x(2x + 5) - 2(2x + 5) =$$(x - 2)(2x + 5)$

  1. Final simplified fraction: $\frac{(x + 4)(x - 4)}{(x - 2)(2x + 5)}$
• What is the derivative of $x^3?$


  $3x^2$
• Find $f(-3)$ if $f(x) =$$x^3 - 2?$


  • Substitute $x =$$-3$ into the function:

  $f(-3) =$$(-3)^3 - 2 =$$-27 - 2 =$$-29$
• How to calculate the value of $f(a)$ if $f(x) =$$5 +$$4x$ and $f(a) =$$73?$


  1. Substitute $f(a)$ into the function:

  $5 +$$4a =$$73$

  1. Solve for $a:$

  $4a =$$73 - 5 =$$68$

  $a =$$68 / 4 =$$17$
• What is $1.08$ as a percentage?


  $108\%$
• A flat's value increased by 8% to £94,500; how much did Nadim pay for it?

  • £94,500 represents $108\%$ of the original value
  • To find the original value, divide by $1.08:$

  $94500 \div 1.08 =$$87,500$

  • Original price: £87,500
• Tommy pays £1610 after a 30% discount; what was the original cost?

  • Discount of 30% means he paid 70% of the original cost
  • Let the original cost be $x:$

  $0.7x =$$1610$

  • Solve for $x:$

  $x =$$1610 / 0.7 =$$2300$

  • Original cost: £2300
• How to calculate a company's annual profit at the end of 2025 if it started at £215,000 and increased by 3% each year?

  1. Multiplier: $1.03$ (100% + 3%)
  2. Apply multiplier over 4 years: $215000 \times 1.03^4$
  3. Calculate: $215000 \times 1.03^4 \approx 241,198.44$
  4. Round to the nearest thousand: £242,000
• How does a car depreciate by 11% in the first year and then by 6% for the next two years?

  • Year 1: Depreciates by 11%, remaining 89% (multiplier $0.89)$
  • Years 2 & 3: Depreciates by 6% each year, remaining 94% each time (multiplier $0.94$ each year)
• A car bought for £2,000 depreciates by 11% in the first year and then by 6% for two years; what is its worth after 3 years?

  1. Year 1: Value = $2000 \times 0.89 =$$1780$
  2. Year 2: Value = $1780 \times 0.94 =$$1669.2$
  3. Year 3: Value = $1669.2 \times 0.94 =$$15690088$
  4. Final worth: £1569.088
• How to express $0.000000288$ in scientific notation?


  • Move the decimal point to the right until you have a number between 1 and 10: $2.88$
  • Count the number of places moved: 7
  • Since the number is less than 1, the exponent is negative: $2.88 \times 10^{-7}$