Practice Numerical Structure in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The underlying patterns, relationships, and algebraic properties—like commutativity and distributivity—that organize numbers into coherent systems.

Numbers aren't random—they have deep structure (primes, factors, operations).

Showing a random 20 of 50 problems.

Example 1

medium
Simplify by combining like structure: 7+3−7+27 + 3 - 7 + 2.

Example 2

medium
Identify which properties (commutative, associative, distributive, identity, inverse) are illustrated by each equation: (a) 3+5=5+33 + 5 = 5 + 3, (b) (2×3)×4=2×(3×4)(2 \times 3) \times 4 = 2 \times (3 \times 4), (c) 7(x+2)=7x+147(x + 2) = 7x + 14, (d) −8+8=0-8 + 8 = 0.

Example 3

medium
Use structure to decide quickly: is 17×417\times 4 even or odd, without multiplying fully?

Example 4

easy
Name the additive inverse of −7-7.

Example 5

easy
Use associativity to group (2+3)+4(2+3)+4 differently.

Example 6

medium
Why can we factor ab+acab+ac back into a(b+c)a(b+c)? Name the property.

Example 7

hard
Simplify: (x+2)(x+3)(x + 2)(x + 3) using structure (FOIL/distribution).

Example 8

medium
Which property justifies rewriting 2×(3×5)2\times(3\times5) as (2×3)×5(2\times3)\times5?

Example 9

easy
Which property is shown by 4+(5+6)=(4+5)+64 + (5 + 6) = (4 + 5) + 6?

Example 10

easy
Use commutativity to rewrite 7+37+3 as an equal sum.

Example 11

medium
What is the additive inverse of 77, and what is the multiplicative inverse of 77?

Example 12

medium
Use the distributive property to compute 7×997\times 99 quickly.

Example 13

easy
What is the multiplicative inverse of 23\frac{2}{3}?

Example 14

hard
What is the multiplicative identity in the integers?

Example 15

challenge
Show structurally that a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).

Example 16

easy
Rewrite 4×54\times 5 using commutativity of multiplication.

Example 17

easy
Compute mentally using distribution: 7×997 \times 99.

Example 18

medium
Does (a−b)−c=a−(b−c)(a-b)-c=a-(b-c) hold? Test with a=10,b=4,c=2a=10,b=4,c=2.

Example 19

hard
Use the distributive property to compute 47×9847 \times 98 mentally, and explain the structural reasoning.

Example 20

easy
Use the distributive property to expand 5(x+7)5(x + 7).
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