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Quantitative Aptitude: Simplification

1. VBODMAS Rule

When solving mathematical expressions, operations must be performed in the following order of precedence:

  1. V - Vinculum (Bar Bracket): Represented by a line over terms, e.g., $\overline{3 - 2}$. Solve this first.
  2. B - Brackets: Solve in order: Parentheses (), Braces {} then Square Brackets [].
  3. O - Of: Equivalent to multiplication but evaluated before division. E.g., $10 \text{ of } 5 = 10 \times 5 = 50$.
  4. D - Division ($/$)
  5. M - Multiplication ($\times$)
  6. A - Addition ($+$)
  7. S - Subtraction ($-$)

2. Laws of Indices & Surds

Laws of Indices

If $a, b$ are real numbers and $m, n$ are rational numbers:

  1. $a^m \times a^n = a^{m+n}$
  2. $\frac{a^m}{a^n} = a^{m-n}$
  3. $(a^m)^n = a^{mn}$
  4. $(ab)^n = a^n \cdot b^n$
  5. $\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$
  6. $a^{-n} = \frac{1}{a^n}$
  7. $a^0 = 1$ (for $a \neq 0$)

Laws of Surds

A surd is an irrational root of a rational number, e.g., $\sqrt{2}, \sqrt[3]{5}$.

  1. $\sqrt[n]{a} = a^{1/n}$
  2. $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
  3. $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$
  4. $(\sqrt[n]{a})^n = a$
  5. $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$

Conjugate and Rationalization

To rationalize a denominator of the form $a \pm \sqrt{b}$, multiply the numerator and denominator by its conjugate $a \mp \sqrt{b}$: $$\frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{(a + \sqrt{b})(a - \sqrt{b})} = \frac{a - \sqrt{b}}{a^2 - b}$$

3. Practice Problems

Problem 1

Simplify the expression: $$108 \div 36 \text{ of } \frac{1}{4} + \frac{2}{5} \times 3\frac{1}{4}$$

Problem 2

Find the value of $x$ if: $$3^{x-1} + 3^{x+1} = 90$$

Problem 3

Simplify by rationalizing: $$\frac{5 + \sqrt{3}}{5 - \sqrt{3}}$$

Problem 4

Evaluate using VBODMAS: $$36 - [18 - {14 - (15 - 4 \div 2 \times 2)}]$$

Problem 5

If $5^{a+b} = 3125$ and $5^{a-b} = 5$, find the value of $a^2 - b^2$.

Problem 6

Simplify the expression: $$\sqrt{5 + 2\sqrt{6}} + \sqrt{5 - 2\sqrt{6}}$$

Problem 7

Evaluate the expression: $$\frac{(2.3)^3 - 0.027}{(2.3)^2 + 0.69 + 0.09}$$

Problem 8

Find the value of $x$ if: $$\sqrt{2^x} = 64$$

Problem 9

Simplify the expression: $$\frac{1}{2} + \frac{1}{2} \div \frac{1}{2} \times \frac{1}{2} - \frac{1}{2}$$

Problem 10

Find the value of $x$ if: $$\frac{x}{1 + \frac{1}{1 - \frac{1}{2}}} = 3$$

4. Step-by-Step Solutions

Solution 1

  1. Convert mixed fraction:
    • $3\frac{1}{4} = \frac{13}{4}$.
    • The expression becomes: $108 \div 36 \text{ of } \frac{1}{4} + \frac{2}{5} \times \frac{13}{4}$.
  2. Evaluate "of" first:
    • $36 \text{ of } \frac{1}{4} = 36 \times \frac{1}{4} = 9$.
    • The expression becomes: $108 \div 9 + \frac{2}{5} \times \frac{13}{4}$.
  3. Evaluate Division:
    • $108 \div 9 = 12$.
    • The expression becomes: $12 + \frac{2}{5} \times \frac{13}{4}$.
  4. Evaluate Multiplication:
    • $\frac{2}{5} \times \frac{13}{4} = \frac{1 \times 13}{5 \times 2} = \frac{13}{10} = 1.3$.
  5. Evaluate Addition:
    • $12 + 1.3 = 13.3$ (or $13\frac{3}{10}$).
  6. Answer: The simplified value is $13.3$.

Solution 2

  1. Factor out $3^{x-1}$ from the left side:
    • Rewrite terms: $3^{x+1} = 3^{x-1} \times 3^2 = 9 \times 3^{x-1}$. $$3^{x-1} + 9 \cdot 3^{x-1} = 90$$ $$3^{x-1} (1 + 9) = 90$$ $$10 \cdot 3^{x-1} = 90$$
  2. Divide by 10: $$3^{x-1} = 9$$
  3. Express 9 as base 3: $$3^{x-1} = 3^2$$
  4. Equate the exponents: $$x - 1 = 2 \implies x = 3$$
  5. Answer: The value of $x$ is 3.

Solution 3

  1. Identify conjugate:
    • Denominator is $5 - \sqrt{3}$.
    • Conjugate is $5 + \sqrt{3}$.
  2. Multiply numerator and denominator by the conjugate: $$\frac{5 + \sqrt{3}}{5 - \sqrt{3}} \times \frac{5 + \sqrt{3}}{5 + \sqrt{3}} = \frac{(5 + \sqrt{3})^2}{5^2 - (\sqrt{3})^2}$$
  3. Expand the terms:
    • Numerator: $(5 + \sqrt{3})^2 = 5^2 + (\sqrt{3})^2 + 2(5)(\sqrt{3}) = 25 + 3 + 10\sqrt{3} = 28 + 10\sqrt{3}$.
    • Denominator: $25 - 3 = 22$.
  4. Simplify the fraction: $$\frac{28 + 10\sqrt{3}}{22} = \frac{2(14 + 5\sqrt{3})}{22} = \frac{14 + 5\sqrt{3}}{11}$$
  5. Answer: The simplified value is $\frac{14 + 5\sqrt{3}}{11}$.

Solution 4

  1. Identify order of operations (VBODMAS):
    • Expression: $36 - [18 - {14 - (15 - 4 \div 2 \times 2)}]$.
  2. Solve the innermost parentheses ():
    • Inside: $15 - 4 \div 2 \times 2$.
    • Perform Division first: $4 \div 2 = 2 \implies 15 - 2 \times 2$.
    • Perform Multiplication next: $2 \times 2 = 4 \implies 15 - 4 = 11$.
  3. Solve the braces {}:
    • Inside: $14 - 11 = 3$.
  4. Solve the square brackets []:
    • Inside: $18 - 3 = 15$.
  5. Solve the final subtraction:
    • $36 - 15 = 21$.
  6. Answer: The value of the expression is 21.

Solution 5

  1. Analyze indices:
    • We are given:
      • $5^{a+b} = 3125$. Since $3125 = 5^5$, we get $a + b = 5$.
      • $5^{a-b} = 5 = 5^1$, we get $a - b = 1$.
  2. Apply algebraic identity:
    • We need to find $a^2 - b^2$.
    • Recall the identity: $a^2 - b^2 = (a + b)(a - b)$.
  3. Substitute the values:
    • $a^2 - b^2 = 5 \times 1 = 5$.
  4. Answer: The value of $a^2 - b^2$ is 5.

Solution 6

  1. Simplify each square root term:
    • Let's express the terms inside the square root as perfect squares.
    • For $\sqrt{5 + 2\sqrt{6}}$:
      • We want to write $5 + 2\sqrt{6}$ in the form $(p + q)^2 = p^2 + q^2 + 2pq$.
      • Let $p = \sqrt{3}$ and $q = \sqrt{2}$.
      • Then $p^2 + q^2 = 3 + 2 = 5$, and $2pq = 2\sqrt{6}$.
      • Thus, $5 + 2\sqrt{6} = (\sqrt{3} + \sqrt{2})^2$.
      • So, $\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}$.
    • Similarly, for $\sqrt{5 - 2\sqrt{6}}$:
      • $5 - 2\sqrt{6} = (\sqrt{3} - \sqrt{2})^2$.
      • So, $\sqrt{5 - 2\sqrt{6}} = \sqrt{3} - \sqrt{2}$ (since $\sqrt{3} > \sqrt{2}$).
  2. Add the simplified terms:
    • $(\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) = 2\sqrt{3}$.
  3. Answer: The simplified value is $2\sqrt{3}$.

Solution 7

  1. Identify algebraic pattern:
    • Look at the numerator: $(2.3)^3 - 0.027$.
    • Since $0.027 = (0.3)^3$, the numerator is of the form $x^3 - y^3$, where $x = 2.3$ and $y = 0.3$.
    • Use identity: $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$.
  2. Expand the terms:
    • $(2.3)^3 - (0.3)^3 = (2.3 - 0.3)((2.3)^2 + 2.3 \times 0.3 + (0.3)^2)$.
    • Summing terms: $2.3 \times 0.3 = 0.69$, and $(0.3)^2 = 0.09$.
    • So the numerator $= 2.0 \times ((2.3)^2 + 0.69 + 0.09)$.
  3. Divide by the denominator:
    • The expression is: $$\frac{2.0 \times ((2.3)^2 + 0.69 + 0.09)}{(2.3)^2 + 0.69 + 0.09} = 2.0$$
  4. Answer: The value is 2 (or 2.0).

Solution 8

  1. Write with exponent notation:
    • $\sqrt{2^x} = (2^x)^{1/2} = 2^{x/2}$.
  2. Convert the RHS to base 2:
    • $64 = 2^6$.
  3. Equate the exponents:
    • $2^{x/2} = 2^6 \implies \frac{x}{2} = 6 \implies x = 12$.
  4. Answer: The value of $x$ is 12.

Solution 9

  1. Apply VBODMAS rules:
    • Expression: $\frac{1}{2} + \frac{1}{2} \div \frac{1}{2} \times \frac{1}{2} - \frac{1}{2}$.
  2. Perform Division:
    • $\frac{1}{2} \div \frac{1}{2} = 1$.
    • The expression becomes: $\frac{1}{2} + 1 \times \frac{1}{2} - \frac{1}{2}$.
  3. Perform Multiplication:
    • $1 \times \frac{1}{2} = \frac{1}{2}$.
    • The expression becomes: $\frac{1}{2} + \frac{1}{2} - \frac{1}{2}$.
  4. Perform Addition and Subtraction:
    • $\frac{1}{2} + \frac{1}{2} = 1$.
    • $1 - \frac{1}{2} = \frac{1}{2}$.
  5. Answer: The simplified value is $\frac{1}{2}$.

Solution 10

  1. Simplify the nested fraction in the denominator:
    • Innermost part: $1 - \frac{1}{2} = \frac{1}{2}$.
    • Next level: $\frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$.
    • Denominator: $1 + 2 = 3$.
  2. Set up the simplified equation:
    • The expression becomes: $\frac{x}{3} = 3$.
  3. Solve for $x$:
    • $x = 3 \times 3 = 9$.
  4. Answer: The value of $x$ is 9.