Hard Level Worksheet Questions

Part A: Subjective Questions

Exponents and powers are fundamental mathematical operations used to express repeated multiplication efficiently. Understanding laws of exponents, negative exponents, and standard form is crucial for algebra, scientific notation, and advanced mathematics.

First, let's explore the fundamental laws of exponents and their applications.

1. Simplify: (2−3 × 42) / 8−1

Step-by-step: Convert to base 2: (2−3 × 24) / 2−3 = 161286432

Excellent! (2−3 × 42) / 8−1 = 24 = 16.

2. Write 0.000027 in standard form. 2.7 × 10−52.7 × 10−427 × 10−60.27 × 10−4

Perfect! 0.000027 = 2.7 × 10−5 (move decimal 5 places right).

3. Evaluate: (30 + 40) × 50 2130

Correct! (1 + 1) × 1 = 2 (any non-zero number to power 0 equals 1).

4. Write the reciprocal of 5−2 without a negative exponent. 25521/255

Great! Reciprocal of 5−2 is 1/5−2 = 1/(1/25) = 25.

5. Simplify: 24 × 2−6 2−2−14222−4

Perfect! 24 × 2−6 = 24−6 = (2)^(-2) = 14.

6. Express 1343 as a power with base 7. 7−3−7373−7−3

Excellent! 343 = 73, so 1343 = 7−3.

7. If ax = a5 and a ≠ 0, find x. 50125

Correct! When bases are equal, exponents must be equal: x = 5.

8. Find the multiplicative inverse of 9−3. 93-72917299−3

Great! Multiplicative inverse of 9−3 is 93 = 729.

9. Write 125 as a power of 5. 53545556

Perfect! 125 = 5 × 5 × 5 = 53.

10. Simplify: 45−2 25161625

Excellent! 45−2 = 542 = 2516.

Drag each expression to the correct exponent law it demonstrates:

Part B: Short Answer Questions (2 Marks Each)

1. Simplify: (3−2 × 9−1) / 27−1

Step 1: Convert to base 3

9−1 = 3−2323−131

27⁻¹ = 3−3333−232

Step 2: Substitute and simplify

(3−2 × 3−2) / 3−3 = 3−2−2+3 = 3−1 = 133199

Perfect! Converting to common base 3 and applying exponent laws gives 1/3.

2. Express 0.00056 in standard form and write the order of magnitude.

Step 1: Standard form

0.00056 = 5.6 × 10−45.6 × 10−356 × 10−50.56 × 10−3

Step 2: Order of magnitude

Order of magnitude = -4-3-5-2

Excellent! 0.00056 = 5.6 × 10−4 with order of magnitude -4.

3. Evaluate: (53 × 5−4 × 52) / (5)^(-3)

Step 1: Simplify numerator

53 × 5−4 × 52 = 53−4+2 = 5150525−1

Step 2: Apply division

51 ÷ 5−3 = 625125255

Great! Using exponent laws: 54 = 625.

4. If x2 × x3 × x−5 = xn, find n.

Step 1: Apply product law

x² × x³ × x⁻⁵ = x0x1x−1x10

Step 2: Find n

Since x0 = xn, therefore n = 01-110

Perfect! The exponents add up to zero, so n = 0.

5. If 5a = 125 and 2b = 8, find a + b.

Step 1: Solve for a

125 = 53, so 5a = 53, therefore a = 3245

Step 2: Solve for b

8 = 23, so 2b = 23, therefore b = 3241

Step 3: Find sum

a + b = 3 + 3 = 6579

Excellent! Both equations give exponent 3, so a + b = 6.

Part C: Long Answer Questions (4 Marks Each)

1. Simplify: (2−4 × 82 × 4−3) / (16−1 × 25) and write as a power of 2.

Step 1: Convert all terms to base 2

82 = 232 = 26252824

4−3 22−3 = 2−62−52−8

16−1 = 24−1 = 2−42−32−52−1

Step 2: Simplify numerator

2−4 × 26 × 2−6 = 2−4+6−6 = 2−42−22022

Step 3: Simplify denominator

2−4 × 25 = 2−4+5 = 212−1292−9

Step 4: Final division

2−4 ÷ 21 = 2−4−1 = 2−52−32325

Outstanding! The final answer is 2−5.

2. Evaluate: 3−2×9327−2−1

Step 1: Convert to base 3

93 = 323 = 36353839

27−2 = 33−2 = 3−63−53−83−9

Step 2: Simplify inside brackets

(3−2 × 36) / 3−6 = 3−2+6−−6 = 3−2+6+6 = 3103831234

Step 3: Apply outer exponent

(3¹⁰)⁻¹ = 3−103103−1031

Perfect! The final answer is 3−10.

3. A bacteria population triples every hour. If initial population is 25, express population after 4 hours.

Step 1: Set up the expression

After 4 hours: Initial × Growthfactor4(Growth factor)4 = 25 × 34334324

Step 2: Calculate 34

34 = 3 × 3 × 3 × 3 = 816427243

Step 3: Calculate final population

25 × 34 = 32 × 81 = 2592256020482304

Excellent! After 4 hours: 25 × 34 = 32 × 81 = 2592 bacteria.

4. Earth diameter ≈ 1.276 × 107 m, Moon diameter ≈ 3.48 × 10⁶ m. Find the ratio.

Step 1: Set up the ratio

Ratio = (1.276 × 107) ÷ (3.48 × 106)

Step 2: Simplify powers of 10

= (1.276 ÷ 3.48) × (10)^(7-8) = (1.276 ÷ 3.48) × 101100x−1102

Step 3: Calculate the ratio

1.276 ÷ 3.48 ≈ 3.670.3736.70.027

Therefore, Earth is approximately 3.67 times larger than Moon.

Perfect! Earth's diameter is about 3.67 times Moon's diameter.

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. Which of the following equals a0 for a ≠ 0?

(a) 0 (b) 1 (c) a (d) –1

Correct! Any non-zero number raised to the power 0 equals 1.

2. Simplify: x3 × x−5

(a) x−2 (b) x8 (c) x2 (d) x−8

Correct! x³ × x⁻⁵ = x³⁻⁵ = x⁻² (add exponents when multiplying).

3. 234 =

(a) 212 (b) 27 (c) 21 (d) 84

Correct! (2³)⁴ = 2³ˣ⁴ = 2¹² (multiply exponents when raising power to power).

4. The standard form of 0.00082 is:

(a) 8.2 × 10−3 (b) 8.2 × 10−4 (c) 82 × 10−4 (d) 0.82 × 10−3

Correct! 0.00082 = 8.2 × 10⁻⁴ (move decimal 4 places to the right).

5. (5)^(-2) =

(a) 25 (b) 1/25 (c) –25 (d) –1/25

Correct! 5⁻² = 1/5² = 1/25 (negative exponent means reciprocal).

Outstanding! You've Mastered Exponents and Powers!

Here's what you accomplished:

• Understanding fundamental laws of exponents (product, quotient, power laws)

• Mastering zero and negative exponents

• Converting numbers to standard form (scientific notation)

• Simplifying complex expressions with multiple bases

• Applying exponent laws to solve equations

• Working with rational exponents and reciprocals

• Solving real-world problems using exponential growth

• Comparing and ordering numbers in exponential form

Your solid foundation in exponents and powers will help you excel in algebra, scientific calculations, and advanced mathematical concepts like logarithms and exponential functions!