Hard Level Worksheet Questions

Part A: Subjective Questions

(1) Find a rational number which is equal to its additive inverse.

A number equals its additive inverse when: x =

This means x =

Awesome! Only 0 equals its additive inverse since 0 = -0.

(2) If a × b = 1, and a = −38, find b. b =

Great job! b = -8/3 is the multiplicative inverse of -3/8.

(3) Write a rational number whose reciprocal is equal to its negative.

The number whose reciprocal is equal to its negative is

Perfect! For x = -1: reciprocal is 1/(-1) = -1

(4) State whether the product of two negative rational numbers is positive or negative. Justify your answer.

Excellent! Negative times negative equals positive.

(5) Write a rational number a such that a + 1a = 0.

(6) Which rational number remains unchanged when added to zero and multiplied by one? Any rational number1-1

That's correct! Zero is the additive identity and one is the multiplicative identity.

(7) Fill in the blank: For any rational number a, a – (–a) =.

Well done! Subtracting a negative is the same as adding.

(8) Is the division of any two rational numbers always a rational number?

Brilliant! Division by zero makes the operation undefined.

(9) Find the multiplicative inverse of –7.

You nailed it! The multiplicative inverse of a is 1/a.

(10) Write the result of: [−56] ÷ [−56]

Perfect! Division of identical non-zero numbers always gives 1.

Drag each example to its correct mathematical property:

(1) Using properties of rational numbers, simplify without actual multiplication: [(59 + (−59)) × (1315)] + (−27)

Simplifying:

Excellent! Using additive inverse property makes this very simple.

(2) Find a rational number x such that: (34) + x = (23)

Solving: x =

Great work! x = -1/12.

(3) A rational number when added to its reciprocal gives 25/12. Find the number.

Let the number be x. Then x + =

Solving: The rational number can be either = or

Perfect! Both 3/4 and 4/3 work (they're reciprocals of each other).

(4) The sum of three rational numbers is –7. Two of them are −32 and 14. Find the third number.

Let third number = x. Then (-3/2) + (1/4) + =

Solving: x =

Outstanding! The third number is -23/4.

(1) Simplify the expression using all relevant properties of rational numbers: {[(−37 + 514) × (43)] – [(25 – 110) × (158)]}

Solving:

Excellent work on this complex multi-step calculation!

(2) Let a = −23, b = 49, and c = −35. Verify whether a × (b + c) = a × b + a × c.

LHS: a × (b + c) =

RHS: a × b + a × c =

Since LHS =≠ RHS, the distributiveassociative property is verified!

Perfect verification of the distributive property!

(3) A student claims that [(−45) ÷ (−23)] ÷ (310) = [(−45) ÷ {(−23) ÷ (310)}]. Verify whether this is correct. Justify your answer with full steps.

LHS: [(−45) ÷ (−23)] ÷ (310) =

RHS: [(−45) ÷ {(−23) ÷ (310)}] =

Since LHS ≠= RHS. Division is not associative.

Excellent demonstration that division is not associative!

(4) Evaluate: {[38 + (−12)] ÷ [(−916) + (58)]} × {[(−14) + 316] ÷ (−716)}

Solving we get

Fantastic work on this complex nested expression!

Part B: Objective Questions

Test your understanding with these challenging multiple choice questions:

1. Which of the following is NOT closed under division in rational numbers?

(a) (34) ÷ (12) (b) (−57) ÷ (23) (c) 0 ÷ (38) (d) (56) ÷ 0

Excellent! Division by zero is undefined, so (56) ÷ 0 is not valid.

2. If (a × b) = 1, then b is the:

(a) Additive inverse of a (b) Multiplicative identity (c) Multiplicative inverse of a (d) Reciprocal of 1

Perfect! If a × b = 1, then b is the multiplicative inverse (reciprocal) of a.

3. Which of the following expressions simplifies to 0?

(a) (34) + (−34) (b) (23) × (32) (c) (56) ÷ (56) (d) (−78) – (78)

Great! A number plus its additive inverse equals zero.

4. A rational number is equal to its reciprocal. The number must be:

(a) 1 (b) –1 (c) 2 (d) Either a or b

Brilliant! Both 1 and -1 equal their own reciprocals: 1/1=1 and 1/(-1)=-1.

5. The simplified value of [(−45) + (65)] × (32) is:

(a) 25 (b) 35 (c) 65 (d) 1.2

Fantastic! [(−45) + (65)] × (32) = (25) × (32) = 610 = 35.

🎉 Exceptional Achievement! You've Mastered Hard-Level Rational Numbers!

Here's what you conquered:

• Advanced rational number properties and special cases
• Complex algebraic relationships (additive/multiplicative inverses)
• Understanding when operations are undefined or have no solutions
• Advanced property verification (distributive, associative, commutative)
• Complex multi-step expressions with nested operations
• Deep understanding of closure properties and their exceptions
• Abstract mathematical reasoning with rational number systems

Your advanced rational number skills prepare you for abstract algebra, advanced mathematics, and rigorous mathematical proof techniques!