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Numerical Aptitude Syllabus / Module 14 · Logs, Surds & Indices / Week 14

Week 14: Logarithms, Surds & Indices

Three related toolkits that make questions with big exponents, ugly square roots, or "solve for x in the exponent" collapse into a few clean lines. None of them are new math — they're a small set of rules, worth having automatic rather than re-derived every time.

Module 14 of 20 Week 14 of 20 ~2–3 Hours Practice Problems Included

By the end of this week, you'll be able to

  • Apply the laws of indices to simplify expressions with exponents
  • Simplify surds and rationalize a denominator quickly
  • Apply logarithm rules to solve for an unknown exponent
1 Laws of Indices 2 Simplifying Surds 3 Rationalizing Denominators 4 Logarithm Rules 5 Practice Problems 6 Knowledge Check

1. Laws of Indices

Six rules cover almost every exponent manipulation this course will ask for — all of them consistent extensions of what "repeated multiplication" already means.

the six core rules
a^m x a^n = a^(m+n)          (multiplying: ADD exponents)
a^m / a^n = a^(m-n)          (dividing: SUBTRACT exponents)
(a^m)^n   = a^(mxn)          (power of a power: MULTIPLY exponents)
a^0       = 1                (any nonzero base to the 0 power)
a^(-n)    = 1/a^n            (negative exponent = reciprocal)
a^(1/n)   = n-th root of a   (fractional exponent = a root)

Worked: simplify (2^3 x 2^5) / 2^4
  = 2^(3+5-4) = 2^4 = 16

The fractional-exponent rule is the bridge to Section 2 — a^(1/2) IS √a, not a separate related idea, which is exactly why surds and indices are taught as one connected topic rather than two.

2. Simplifying Surds

A surd is an irrational root left in root form (like √12) rather than converted to a decimal. Simplifying means pulling out the largest perfect-square factor.

simplifying by extracting perfect squares
Simplify sqrt(72)

72 = 36 x 2   (36 is the largest perfect-square factor of 72)
sqrt(72) = sqrt(36 x 2) = sqrt(36) x sqrt(2) = 6 x sqrt(2)
         = 6 sqrt(2)

Combining surds -- only LIKE surds add directly (same root part):
  3 sqrt(2) + 5 sqrt(2) = 8 sqrt(2)   -- like sqrt(2), combine
  3 sqrt(2) + 5 sqrt(3) = CANNOT combine further -- different
                           root parts, stays as two separate terms

This mirrors combining "like terms" in ordinary algebra (3x + 5x = 8x, but 3x + 5y stays separate) — surds with the same root part behave exactly like a variable, and surds with different root parts are as unrelated as two different variables.

3. Rationalizing Denominators

A fraction with a surd in the denominator (like 1/√2) is conventionally rewritten with a rational (non-surd) denominator, by multiplying top and bottom by the surd itself.

rationalizing a simple surd denominator
1/sqrt(2)  ->  multiply top and bottom by sqrt(2):

  1/sqrt(2) x sqrt(2)/sqrt(2) = sqrt(2)/2

(Multiplying by sqrt(2)/sqrt(2) = 1 doesn't change the
VALUE, only the form -- sqrt(2) x sqrt(2) = 2, a rational
number, in the denominator.)
rationalizing a binomial surd denominator (using the conjugate)
1/(3 + sqrt(2))  ->  multiply by the CONJUGATE (3 - sqrt(2))

  1/(3+sqrt(2)) x (3-sqrt(2))/(3-sqrt(2))
    = (3-sqrt(2)) / (3^2 - (sqrt(2))^2)
    = (3-sqrt(2)) / (9-2)
    = (3-sqrt(2)) / 7

The conjugate trick uses (a+b)(a-b)=a^2-b^2 -- the surd
term SQUARES AWAY entirely, leaving a rational denominator.

4. Logarithm Rules

A logarithm answers "what exponent gives this result?" — it's the inverse operation of exponentiation, and its rules are Section 1's index laws, translated.

what a logarithm means
log_b(x) = y   MEANS THE SAME THING AS   b^y = x

Worked: log_2(8) = ? -> "2 to what power gives 8?" -> 3
  (check: 2^3 = 8, correct)
the core log rules
log(a x b) = log(a) + log(b)      (product -> SUM of logs)
log(a / b) = log(a) - log(b)      (quotient -> DIFFERENCE)
log(a^n)   = n x log(a)           (power -> MULTIPLY out front)
log_b(b)   = 1
log_b(1)   = 0

Worked: solve for x in 3^x = 81
  Take log of both sides: log(3^x) = log(81)
  x log(3) = log(81)
  x = log(81)/log(3)
  -- or, faster, recognize 81 = 3^4 directly -> x=4

The "sum of logs for a product" rule is exactly why logarithms were historically used to turn multiplication into addition (via log tables) — a detail worth knowing conceptually, since it's precisely why log(a×b) = log(a) + log(b) and not some other combination.

Try recognizing the exact power before reaching for log rules

Many test-level exponential equations (like 3^x=81) are faster solved by recognizing the base's power directly (81=3^4) than by formally taking logs of both sides — save the log rules for cases where the numbers genuinely don't share a clean common base, which is a smaller fraction of questions than it might first appear.

5. Practice Problems

Practice

30 problems, no calculator — Easy, Medium & Tough

Try to recognize clean powers before reaching for logarithm rules. Easy problems should take under 30 seconds each; Medium under 60 seconds; Tough under 90 seconds.

Easy (1–10)

  1. Simplify: 2³ × 2⁴.
  2. Simplify: 3⁵ / 3².
  3. Simplify: (2²)³.
  4. Evaluate: 5⁰.
  5. Evaluate: 2⁻³.
  6. Simplify: √16.
  7. Simplify √50 (leave in simplest surd form).
  8. Evaluate: log₂8.
  9. Evaluate: log₁₀100.
  10. Simplify: √4 × √9.
Easy — Answers

1) 2⁷=128. 2) 3³=27. 3) 2⁶=64. 4) 1. 5) 1/8. 6) 4. 7) 5√2. 8) 3 (2³=8). 9) 2. 10) 2×3=6.

Medium (11–20)

  1. Simplify: (3⁴ × 3²) / 3³.
  2. Simplify √72.
  3. Simplify: 2√8 + 3√2.
  4. Rationalize: 1/√3.
  5. Evaluate: log₃81.
  6. Solve for x: 2ˣ = 32.
  7. Simplify: log(20) + log(5) using the log rules.
  8. Simplify: (5³)² / 5⁴.
  9. Simplify: √3 × √12.
  10. Evaluate: log₅125.
Medium — Answers

11) 3⁽⁴⁺²⁻³⁾=3³=27. 12) 6√2. 13) 2√8=4√2; 4√2+3√2=7√2. 14) √3/3. 15) 3⁴=81 → 4. 16) 2⁵=32 → x=5. 17) log(20×5)=log(100)=2. 18) 5⁽⁶⁻⁴⁾=5²=25. 19) √36=6. 20) 5³=125 → 3.

Tough (21–30)

  1. Simplify: √48 + √27 - √12.
  2. Rationalize: 1/(4 + √2).
  3. Solve for x: 3ˣ = 243.
  4. Simplify: log(45) - log(9).
  5. Simplify: (2⁵ × 2⁻²) / 2⁰.
  6. Find the value of x if log₂(x) = 5.
  7. Simplify: (√5 + √3)(√5 - √3).
  8. Solve for x: 5^(2x-1) = 125.
  9. Simplify: log₂64 + log₂4 - log₂8.
  10. Rationalize and simplify: (3+√2)/(3-√2).
Tough — Answers

21) √48=4√3, √27=3√3, √12=2√3; 4√3+3√3-2√3=5√3. 22) Multiply by (4-√2)/(4-√2): (4-√2)/14. 23) 3⁵=243 → x=5. 24) log(45/9)=log(5). 25) 2⁽⁵⁻²⁻⁰⁾=2³=8. 26) x=2⁵=32. 27) (√5)²-(√3)²=5-3=2. 28) 5³=125 → 2x-1=3 → x=2. 29) 6+2-3=5. 30) Multiply by (3+√2)/(3+√2): (9+6√2+2)/(9-2)=(11+6√2)/7.

6. Knowledge Check

Four quick questions. Expand each to check your answer.

Q1

Why does a^(1/2) equal √a — what connects a fractional exponent to a root?

For the exponent rules to stay consistent, a^(1/2) × a^(1/2) must equal a^(1/2+1/2) = a^1 = a — and the number that, multiplied by itself, gives a is exactly the definition of √a. So a^(1/2) has to equal √a for the "add exponents when multiplying" rule to keep holding true for fractional exponents too.

Q2

Why can 3√2 and 5√2 be combined into 8√2, but 3√2 and 5√3 cannot be combined further?

√2 acts as a single, consistent "unit" being counted, exactly like a variable — 3 of that unit plus 5 of the same unit is 8 of it. √2 and √3 are different, unrelated irrational values, the same way x and y are different variables that can't be combined into a single term by simple addition.

Q3

Why does multiplying by the conjugate (3-√2) eliminate the surd from a denominator like (3+√2), when multiplying by √2/√2 alone wouldn't?

The conjugate pair (a+b)(a-b) expands to a²-b², which eliminates the cross terms that would otherwise still contain the surd — squaring √2 turns it into the rational number 2. Multiplying just by √2/√2 only works when the denominator IS a lone surd; for a binomial denominator like (3+√2), only the conjugate trick produces the needed cancellation.

Q4

Why does log(a×b) equal log(a) + log(b), rather than log(a) × log(b)?

A logarithm is fundamentally an exponent, and multiplying two numbers with the same base means ADDING their exponents (Section 1's first index law) — since log(a) and log(b) are exactly those exponents, the exponent of the product a×b is their sum, which is what the logarithm of that product reports.

← Back to Week 13: Probability Up next Week 15: Geometry & Mensuration →