Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

In this hard level, we'll tackle negative exponents, complex operations, and advanced applications.

Let's master the most challenging concepts in exponents!

1. Simplify: (2³ × 2² × 2⁴).

2³ × 2² × 2⁴ = 2924

Perfect! Add all exponents: 3 + 2 + 4 = 9.

2. Simplify: (x⁵ ÷ x³ × x²).

Work left to right: x⁵ ÷ x³ = x28

x² × x² = x40

Excellent! Division first, then multiplication.

3. Write 0.001 as a power of 10.

0.001 = 1/1000100

1000 = 1034

So 0.001 = 10 to the power negative 33

Correct! Negative exponent for fractions: 1/10³ = 10⁻³.

4. Simplify: (3²)³.

(3²)³ = 365

= 729243

Perfect! Multiply exponents: 2 × 3 = 6.

5. Write reciprocal of 4⁻².

Reciprocal of 4⁻² = 4²1/16

= 164

Excellent! Reciprocal changes sign of exponent.

6. Simplify: (a⁻³ × a⁵).

a⁻³ × a⁵ = a28

Correct! Add exponents: -3 + 5 = 2.

7. Simplify: (2⁴ ÷ 2⁻²).

2⁴ ÷ 2⁻² = 262

Perfect! 4 - (-2) = 4 + 2 = 6.

8. Write 1/8 as a power of 2.

8 = 234

1/8 = 2 to the power negative 33

Excellent! 1/2³ = 2⁻³.

9. Simplify: (5³)⁻¹.

(5³)⁻¹ = 5 to the power negative 33

= 1/125125

Correct! Negative exponent means reciprocal.

10. Simplify: [(3²)³ ÷ 3⁴].

(3²)³ = 365

3⁶ ÷ 3⁴ = 3210

= 981

Perfect! Multiple operations combined.

Drag each expression to its simplified form:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Simplify: (a⁴ × a⁻² × a³).

a⁴ × a⁻² × a³ = a51

Perfect! 4 + (-2) + 3 = 5.

2. Simplify: (x⁵ ÷ x⁻²).

x⁵ ÷ x⁻² = x73

Excellent! 5 - (-2) = 5 + 2 = 7.

3. Simplify: (2³ × 3² ÷ 2²).

Step 1: 2³ ÷ 2² = 215

Step 2: 2¹ × 3² = 21 × 93

= 1811

Great! Combine same bases first.

4. Simplify: (4⁵ ÷ 4⁻³).

4⁵ ÷ 4⁻³ = 482

Perfect! 5 - (-3) = 8.

5. Simplify: (a⁻³ × a⁵ ÷ a²).

Step 1: a⁻³ × a⁵ = a28

Step 2: a² ÷ a² = a04

= 10

Excellent! a⁰ = 1.

6. Simplify: (10⁻² × 10³).

10⁻² × 10³ = 1015

= 10100

Correct! -2 + 3 = 1.

7. Simplify: [(x²)³ × x⁻²].

(x²)³ = x65

x⁶ × x⁻² = x48

Perfect! 6 + (-2) = 4.

8. Simplify: (5⁴ ÷ 5⁻¹).

5⁴ ÷ 5⁻¹ = 553

= 3125625

Excellent! 4 - (-1) = 5.

9. Simplify: [(2⁴)³ ÷ 2⁶].

(2⁴)³ = 2127

2¹² ÷ 2⁶ = 2618

= 6432

Great! Power of power, then division.

10. Simplify: [(x³ × y⁻²) ÷ (x⁻² × y³)].

For x: x³ ÷ x⁻² = x51

For y: y⁻² ÷ y³ = y to the power negative 51

Result: x⁵ divided by y⁵

Perfect! Handle each variable separately.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Simplify using laws of exponents: [(a³ × b⁻²) ÷ (a⁻¹ × b³)].

Step 1: Separate by variables

For a: a³ ÷ a⁻¹ = a42

For b: b⁻² ÷ b³ = b to the power negative 51

Step 2: Combine

= a⁴ divided by b⁵

Excellent! 3 - (-1) = 4 and -2 - 3 = -5.

2. Simplify: [(2⁵ × 3²) ÷ (6³)].

Step 1: Expand 6³

6³ = (2 × 3)³ = 2³ × 3³

Step 2: Rewrite

(2⁵ × 3²) ÷ (2³ × 3³)

Step 3: Simplify each base

For 2: 2⁵ ÷ 2³ = 228

For 3: 3² ÷ 3³ = 3 to the power negative 15

Step 4: Combine

2² × 3⁻¹ = 42 × 1/33

= 4/312

Perfect! Complex base splitting and simplification.

3. Simplify: [(x⁴ × y⁻³) ÷ (x⁻² × y²)].

For x:

x⁴ ÷ x⁻² = x62

For y:

y⁻³ ÷ y² = y to the power negative 5-1

Result:

= x⁶ divided by y⁵

Excellent! Multi-variable with negative exponents.

4. Simplify: [(5⁴ × 2³) ÷ (10²)].

Step 1: Expand 10²

10² = (5 × 2)² = 5² × 2²

Step 2: Rewrite

(5⁴ × 2³) ÷ (5² × 2²)

Step 3: Simplify

For 5: 5⁴ ÷ 5² = 526

For 2: 2³ ÷ 2² = 215

Step 4: Calculate

5² × 2¹ = 255 × 21

= 5027

Perfect! Breaking down composite bases.

5. Simplify: [(3² × 2⁻¹ × 6³) ÷ 3³].

Step 1: Expand 6³

6³ = (2 × 3)³ = 2³ × 3³

Step 2: Rewrite

(3² × 2⁻¹ × 2³ × 3³) ÷ 3³

Step 3: Group same bases

For 3: (3² × 3³) ÷ 3³ = 325

For 2: 2⁻¹ × 2³ = 224

Step 4: Calculate

3² × 2² = 93 × 42

= 3613

Amazing! Complex multi-base problem solved.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. (4⁵ ÷ 4⁻²) =

(a) 4⁷ (b) 4⁻⁷ (c) 4³ (d) 4²

Correct! 5 – (–2) = 5 + 2 = 7.

7. (3²)⁴ =

(a) 3⁶ (b) 3⁸ (c) 3⁴ (d) 3²

Perfect! Multiply exponents: 2 × 4 = 8.

8. The reciprocal of 2⁻³ is:

(a) 2³ (b) 2⁻³ (c) 1/2³ (d) 2⁶

Correct! Reciprocal changes sign: 1/(2⁻³) = 2³.

9. 5⁻² =

(a) 1/25 (b) 25 (c) 1/5 (d) 10

Perfect! 5⁻² = 1/5² = 1/25.

10. Simplify: [(2⁴ × 2⁻³) ÷ 2⁻²] =

(a) 2³ (b) 2² (c) 2⁵ (d) 2⁻³

Excellent! (4 – 3) – (–2) = 1 + 2 = 3.

🏆 Outstanding Achievement! You've Mastered Advanced Powers and Exponents!

Here's what you've conquered at the hard level:

• Negative Exponents - Key Concept:

  Definition:

  • a⁻ⁿ = 1/aⁿ
  • Negative exponent means reciprocal

  Examples:

  • 2⁻³ = 1/2³ = 1/8
  • 5⁻² = 1/5² = 1/25
  • 10⁻¹ = 1/10 = 0.1

  Reciprocal relationship:

  • Reciprocal of aⁿ = a⁻ⁿ
  • Reciprocal of a⁻ⁿ = aⁿ
  • Example: Reciprocal of 2⁻³ = 2³ = 8

• Operations with Negative Exponents:

  Multiplication:

  • a⁻ᵐ × aⁿ = aⁿ⁻ᵐ
  • Example: x⁻² × x⁵ = x³
  • Add exponents (watch signs!)

  Division:

  • aᵐ ÷ a⁻ⁿ = aᵐ⁺ⁿ
  • Example: 5⁴ ÷ 5⁻² = 5⁶
  • Subtracting negative = adding positive

  Power of negative:

  • (a⁻ᵐ)ⁿ = a⁻ᵐⁿ
  • Example: (2⁻³)² = 2⁻⁶

• Decimal Representation:

  Negative powers of 10:

  • 10⁻¹ = 0.1
  • 10⁻² = 0.01
  • 10⁻³ = 0.001
  • 10⁻⁴ = 0.0001

  Converting decimals:

  • 0.1 = 1/10 = 10⁻¹
  • 0.01 = 1/100 = 10⁻²
  • 0.001 = 1/1000 = 10⁻³

• Complex Multi-Variable Expressions:

  Strategy:

  1. Separate variables
  2. Apply laws to each variable
  3. Combine results

  Example:

  • (x³ × y⁻²) ÷ (x⁻¹ × y²)
  • For x: x³ ÷ x⁻¹ = x⁴
  • For y: y⁻² ÷ y² = y⁻⁴
  • Result: x⁴/y⁴ = (x/y)⁴

• Composite Base Simplification:

  Breaking down composite bases:

  • 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ
  • 10ⁿ = (2 × 5)ⁿ = 2ⁿ × 5ⁿ

  Example:

  • (2³ × 5²) ÷ 10²
  • 10² = 2² × 5²
  • = (2³ × 5²) ÷ (2² × 5²)
  • = 2¹ × 5⁰ = 2 × 1 = 2

• Complete Laws Summary:

  1. aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents)
  2. aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract exponents)
  3. (aᵐ)ⁿ = aᵐⁿ (multiply exponents)
  4. (ab)ⁿ = aⁿbⁿ (distribute power)
  5. (a/b)ⁿ = aⁿ/bⁿ (distribute power)
  6. a⁻ⁿ = 1/aⁿ (negative = reciprocal)
  7. a⁰ = 1 (zero power = 1)
  8. a¹ = a (power of 1)

• Problem-Solving Framework:

  Step 1: Analyze

  • Identify all bases
  • Note negative exponents
  • Plan order of operations

  Step 2: Simplify

  • Group same bases
  • Apply appropriate laws
  • Handle negatives carefully

  Step 3: Calculate

  • Expand composite bases if needed
  • Perform operations
  • Convert to positive exponents if required

  Step 4: Verify

  • Check signs
  • Ensure fully simplified
  • Test with simple values if unsure

• Advanced Patterns:

  • a⁻ⁿ × aⁿ = a⁰ = 1
  • (aⁿ)⁻¹ = a⁻ⁿ
  • (a⁻¹)⁻¹ = a
  • (a/b)⁻ⁿ = (b/a)ⁿ
  • a⁻ᵐ/b⁻ⁿ = bⁿ/aᵐ

• Critical Sign Management:

  • Adding negative: m + (–n) = m – n
  • Subtracting negative: m – (–n) = m + n
  • Multiplying negatives: (–m)(–n) = mn
  • Always use parentheses for clarity

• Real-World Applications:

  • Scientific notation: 3.2 × 10⁻⁵
  • Microscopic measurements: bacteria size
  • Computer memory: bytes (2¹⁰, 2²⁰, etc.)
  • Chemistry: pH scale, concentration
  • Physics: wavelengths, atomic sizes
  • Finance: compound interest calculations

• Common Advanced Mistakes:

  • Forgetting to change sign when dividing by negative
  • Adding instead of multiplying for power of power
  • Not distributing power to all factors
  • Confusing a⁻ⁿ with –aⁿ (very different!)
  • Not simplifying composite bases
  • Losing track of multiple negative signs

• Mastery Checklist: ✓ Understand negative exponents as reciprocals ✓ Apply all laws fluently with any exponents ✓ Handle multi-variable expressions ✓ Break down composite bases ✓ Convert between forms (decimal, fraction, exponential) ✓ Solve complex multi-step problems ✓ Verify answers make logical sense

Mastering exponents unlocks advanced mathematics, science, and engineering - you now have the power!

Remember: Practice with increasingly complex problems to maintain and sharpen your skills!

Key to success: Careful sign management, systematic approach, and understanding not just memorizing!