Exponents & Powers — Study Notes

Overview

Exponents and powers form a core computational skill tested in the UPSSSC PET Elementary Arithmetic section. You will face direct calculation questions (simplify 2⁵ × 2³), law-of-exponent application problems, and scientific notation conversions. This topic bridges pure arithmetic and higher algebra, so mastery here also strengthens your aptitude for data interpretation and percentage problems where large numbers appear.

Expect 2–4 questions directly or indirectly involving exponents in the 100-question PET paper. Questions are straightforward but designed to catch careless errors in sign handling, zero exponents, and negative bases. The key is to memorise the five fundamental laws and practise until application becomes automatic. Scientific notation questions test your ability to express very large or very small numbers compactly — a skill useful in General Science and Graph Interpretation sections as well.

Focus on speed. Most exponent questions should take 30–45 seconds. Slow manual multiplication is a trap; always apply laws to simplify before computing.

Key Concepts

• **Exponent (or Power or Index)**: In the expression aⁿ, 'a' is the base and 'n' is the exponent. It means 'a' multiplied by itself 'n' times. For example, 3⁴ = 3 × 3 × 3 × 3 = 81.

• **Positive Integer Exponent**: The exponent tells how many times the base appears in the product. 5³ = 5 × 5 × 5 = 125.

• **Zero Exponent**: Any nonzero number raised to the power zero equals 1. That is, a⁰ = 1 (where a ≠ 0). Example: 7⁰ = 1, (-3)⁰ = 1.

• **Negative Exponent**: A negative exponent indicates the reciprocal. a⁻ⁿ = 1/aⁿ. Example: 2⁻³ = 1/2³ = 1/8.

• **Fractional Exponent (Mentioned for Completeness)**: In PET, stick to integer exponents. If a question involves roots, convert: √a = a^(1/2).

• **Scientific Notation**: A convenient way to write very large or very small numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Example: 45000 = 4.5 × 10⁴; 0.00032 = 3.2 × 10⁻⁴.

• **Order of Magnitude**: The exponent on 10 in scientific notation tells you the scale. 10⁶ is one million, 10⁹ is one billion (short scale), 10⁻³ is one-thousandth.

• **Base 10 vs Other Bases**: Most PET questions use base 10 for scientific notation and bases 2, 3, 5 for law-of-exponent problems. Rarely you may see base e (but unlikely at PET level).

Formulas / Key Facts

**The Five Fundamental Laws of Exponents:**

1. **Product of Powers (Same Base)**: aᵐ × aⁿ = a^(m+n). When multiplying like bases, add exponents. Example: 2³ × 2⁵ = 2⁸.

2. **Quotient of Powers (Same Base)**: aᵐ / aⁿ = a^(m−n). When dividing like bases, subtract exponents. Example: 5⁷ / 5⁴ = 5³.

3. **Power of a Power**: (aᵐ)ⁿ = a^(m×n). When raising a power to another power, multiply exponents. Example: (3²)⁴ = 3⁸.

4. **Power of a Product**: (a × b)ⁿ = aⁿ × bⁿ. Distribute the exponent over each factor. Example: (2 × 5)³ = 2³ × 5³ = 8 × 125 = 1000.

5. **Power of a Quotient**: (a/b)ⁿ = aⁿ / bⁿ. Distribute the exponent over numerator and denominator. Example: (3/4)² = 3² / 4² = 9/16.

**Special Cases:**

• **a¹ = a**: Any number to the power 1 is itself.
• **a⁰ = 1** (a ≠ 0): Any nonzero base to the power zero is 1.
• **1ⁿ = 1**: 1 raised to any power is 1.
• **0ⁿ = 0** (n > 0): Zero raised to any positive power is zero.
• **a⁻ⁿ = 1/aⁿ**: Negative exponent means reciprocal.

**Powers of 2 (Memorise up to 2¹⁰):** 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024.

**Powers of 3, 5, 10 (Useful):** 3²=9, 3³=27, 3⁴=81, 3⁵=243; 5²=25, 5³=125; 10²=100, 10³=1000, 10⁴=10000.

**Scientific Notation Rules:**

• To convert a large number: move decimal left until one nonzero digit remains left of decimal; count moves = positive exponent on 10.
• To convert a small decimal: move decimal right until one nonzero digit remains left of decimal; count moves = negative exponent on 10.

Worked Examples

**Example 1: Simplify 3⁴ × 3² / 3³**

Step 1: Apply product rule to numerator: 3⁴ × 3² = 3^(4+2) = 3⁶. Step 2: Now divide: 3⁶ / 3³ = 3^(6−3) = 3³. Step 3: Compute 3³ = 27. **Answer: 27**

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**Example 2: Evaluate (2³)² × 2⁻⁴**

Step 1: Apply power-of-a-power: (2³)² = 2^(3×2) = 2⁶. Step 2: Multiply: 2⁶ × 2⁻⁴ = 2^(6−4) = 2². Step 3: Compute 2² = 4. **Answer: 4**

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**Example 3: Express 0.000052 in scientific notation**

Step 1: Move decimal 5 places to the right to get 5.2 (one nonzero digit before decimal). Step 2: Since we moved right, exponent is negative: 5.2 × 10⁻⁵. **Answer: 5.2 × 10⁻⁵**

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**Example 4: Simplify [(5²)³ × 5⁻⁴] / 5³**

Step 1: Simplify numerator: (5²)³ = 5⁶, then 5⁶ × 5⁻⁴ = 5^(6−4) = 5². Step 2: Divide: 5² / 5³ = 5^(2−3) = 5⁻¹. Step 3: Convert negative exponent: 5⁻¹ = 1/5 = 0.2. **Answer: 1/5 or 0.2**

Common Mistakes

**Mistake 1: Adding exponents when bases are different.** Wrong: 2³ × 3² = 6⁵. Correct: You cannot combine; compute separately: 2³ = 8, 3² = 9, so 8 × 9 = 72. Laws apply only to *same base*.

**Mistake 2: Confusing (aᵐ)ⁿ with aᵐ × aⁿ.** Wrong: (2³)² = 2⁵. Correct: (2³)² = 2^(3×2) = 2⁶ = 64, not 2⁵ = 32. Multiply exponents when power-of-a-power, add when multiplying same base.

**Mistake 3: Treating a⁰ as 0 instead of 1.** Wrong: 5⁰ = 0. Correct: 5⁰ = 1. Any nonzero base to power zero equals 1. Remember this by the quotient rule: aⁿ / aⁿ = a^(n−n) = a⁰ = 1.

**Mistake 4: Sign errors with negative bases.** Wrong: (−2)⁴ = −16. Correct: (−2)⁴ = (−2)×(−2)×(−2)×(−2) = +16. Even power makes positive; odd power keeps negative. Contrast: −2⁴ (without parentheses) = −(2⁴) = −16.

**Mistake 5: Misplacing the decimal in scientific notation.** Wrong: 3400 = 34 × 10². Correct: 3400 = 3.4 × 10³. The coefficient must be between 1 and 10. Always leave exactly one nonzero digit before the decimal point.

Quick Reference

• **Product Rule**: aᵐ × aⁿ = a^(m+n) — add exponents (same base).
• **Quotient Rule**: aᵐ / aⁿ = a^(m−n) — subtract exponents (same base).
• **Power Rule**: (aᵐ)ⁿ = a^(m×n) — multiply exponents.
• **Zero Exponent**: a⁰ = 1 (a ≠ 0).
• **Negative Exponent**: a⁻ⁿ = 1/aⁿ — flip to reciprocal.
• **Scientific Notation Format**: a × 10ⁿ where 1 ≤ a < 10.
• **Powers of 10**: 10³ = 1000 (kilo), 10⁶ = 1 million (mega), 10⁻³ = 0.001 (milli), 10⁻⁶ = 0.000001 (micro).
• **Speed Trick**: Memorise 2¹ to 2¹⁰ and 3² to 3⁵ to avoid recalculation in exam.

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**Final Tip**: Practice 20–30 mixed problems covering all five laws and scientific notation. Most errors come from rushing; slow down by 5 seconds per question to double-check base and sign. Accuracy beats speed when each mark counts.