Cube Tricks for Fast and Accurate Cube Calculation

Cube Tricks provide quick methods for finding the cubes of numbers without long multiplication. This page explains algebraic cube formulas, numbers near a base such as 10 or 100, two-digit number methods, powers of numbers ending in zero, and sign rules with clear calculations.

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What Are Cube Tricks?

Cube tricks are shortcut methods based on algebraic identities and place-value rules used to calculate n³ quickly. The main identities are (a + b)³ = a³ + 3a²b + 3ab² + b³ and (a − b)³ = a³ − 3a²b + 3ab² − b³.

A number can be expressed as a convenient base plus or minus a small value. For example, 12³ = (10 + 2)³ = 10³ + 3(10²)(2) + 3(10)(2²) + 2³ = 1000 + 600 + 120 + 8 = 1728. The cube of a negative number is negative, while the cube of zero is zero.

Cube Tricks Formula & Tricks

Important Formulas

Cube of a sum
(a + b)³ = a³ + 3a²b + 3ab² + b³

Use this when a number is written as a convenient base plus a smaller part, such as 12 = 10 + 2.

Cube of a difference
(a − b)³ = a³ − 3a²b + 3ab² − b³

Use this when a number is slightly less than a convenient base, such as 97 = 100 − 3.

Two-digit cube formula
(10a + b)³ = 1000a³ + 300a²b + 30ab² + b³

For a two-digit number with tens digit a and units digit b, calculate the four terms and add them.

Cube of a product
(ab)³ = a³b³

Cube each factor separately. For example, 30³ = 3³ × 10³ = 27 × 1000 = 27000.

Quick Tricks

Choose a nearby base

Write the number as B + x or B − x, where B is usually 10, 100 or 1000. This makes the first cube and the remaining products easier to calculate.

Example: 103³ = (100 + 3)³ = 1,000,000 + 90,000 + 2,700 + 27 = 1,092,727.
Use alternating signs for a difference

In (a − b)³, the signs are positive, negative, positive, negative. The middle positive term is 3ab², not negative.

Example: 97³ = (100 − 3)³ = 1,000,000 − 90,000 + 2,700 − 27 = 912,673.
Separate powers of 10

If a number is a product involving 10, cube the non-zero factor and multiply by the corresponding power of 10.

Example: 40³ = (4 × 10)³ = 4³ × 10³ = 64 × 1000 = 64000.
Check the last digit

The last digit of a cube depends only on the last digit of the original number. For example, a number ending in 7 has a cube ending in 3 because 7³ = 343.

Example: 23³ = 12167, which ends in 7 because 3³ = 27. The last digit check confirms the result.

Cube Tricks Concepts

Cube of a Two-Digit Number

For a two-digit number 10a + b, use (10a + b)³ = 1000a³ + 300a²b + 30ab² + b³.

The four terms represent the expansion of (10a + b)³. Calculate each term separately and then add them. For 23³, take a = 2 and b = 3: 1000(8) + 300(4)(3) + 30(2)(9) + 27 = 8000 + 3600 + 540 + 27 = 12167.

Example: 14³ = 1000(1³) + 300(1²)(4) + 30(1)(4²) + 4³ = 1000 + 1200 + 480 + 64 = 2744.

Cubes Near 10, 100 or 1000

For a number close to a base B, write it as B + x or B − x and apply the cube identity.

For a number above the base, use B³ + 3B²x + 3Bx² + x³. For a number below the base, use B³ − 3B²x + 3Bx² − x³. The value x is the difference from the base.

Example: 98³ = (100 − 2)³ = 1,000,000 − 60,000 + 1,200 − 8 = 941192.

Cubes of Numbers Ending in Zero

To cube a number ending in zero, cube the number after removing the zero and append three zeros for each factor of 10.

Since (10a)³ = 1000a³, one trailing zero in the original number contributes three trailing zeros in its cube. Thus, 70³ = 7³ × 1000 = 343000.

Example: 120³ = (12 × 10)³ = 12³ × 10³ = 1728 × 1000 = 1728000.

Sign Rule for Cubes

A positive number has a positive cube, and a negative number has a negative cube.

The rule follows from (-a)³ = (-a)(-a)(-a) = -a³. The product of three negative factors is negative, so (-5)³ = -125. Do not confuse this with an even power such as (-5)² = 25.

Example: (-12)³ = -(12³) = -1728.

Cube Calculation Using Factorisation

When a number can be factored into simple factors, cube each factor and multiply the results.

Use (ab)³ = a³b³. This method is convenient when the factors have known cubes, such as 2, 3, 5, 10 or their multiples.

Example: 15³ = (3 × 5)³ = 3³ × 5³ = 27 × 125 = 3375.

Cube Tricks Video Lessons

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8 Lessons
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Vedic Maths Cube Tricks

Learn practical Vedic Maths techniques for calculating cubes efficiently, with clear methods that help simplify cube calculations in quantitative aptitude.

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Practice Cube Tricks Questions

Practise published questions related to this topic.

1√45 × √5 = ?→ 2198 chips and 330 toffees are to be distributed among maximum number of boys. Each boy gets the same number of chips. Each boy get the same number of toffees. How many chips does each boy get?→ 3The sum of a number and its positive square root is 156. What is the value of the number?→ 4Which of the following will have 25 percent discount?→ 5Given that 119^(0.49) = x , 119^(0.62) = y and x^z = y⁶, then the value of z is close to:→ 61/(6 + √3) + 1/(6 - √3) = ?→ 7What is the greatest length of a measuring rod that can exactly measure three fabric pieces of lengths 7 m 35 cm, 9 m 45 cm, and 10 m 50 cm?→ 8A person sells an article at the profit of 50 percent. Cost price and selling price are increased by 40 percent and 'y' percent respectively. If the new profit percentage is 75 percent, then what is the value of 'y'?→ 9Three numbers are in the ratio of 19 : 7 : 13, and their LCM is 8645. Their HCF is:→ 10A person has a loss of 20 percent on selling price. What will be his loss percentage on the cost price?→

Cube Tricks Quick Quiz

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Quick Revision Notes

Cube Tricks: Quick Revision

Use these identities and rules to calculate cubes accurately.

  • (a + b)³ = a³ + 3a²b + 3ab² + b³.
  • (a − b)³ = a³ − 3a²b + 3ab² − b³.
  • For a number near B, write it as B + x or B − x.
  • The signs in (a − b)³ follow the pattern positive, negative, positive, negative.
  • (ab)³ = a³b³.
  • (10a)³ = 1000a³, so one trailing zero becomes three trailing zeros.
  • The cube of a negative number is negative.
  • Check the final digit using the cube of the original number's units digit.

Cube Tricks FAQs

What is the formula for (a + b)³?

(a + b)³ = a³ + 3a²b + 3ab² + b³.

What is the formula for (a − b)³?

(a − b)³ = a³ − 3a²b + 3ab² − b³.

How can 99³ be calculated quickly?

Write 99 = 100 − 1. Therefore, 99³ = 1,000,000 − 30,000 + 300 − 1 = 970299.

How do you calculate 102³ using a cube shortcut?

102 = 100 + 2, so 102³ = 1,000,000 + 60,000 + 1,200 + 8 = 1,061,208.

What is the cube of a negative number?

The cube of a negative number is negative because (-a)³ = -a³. For example, (-8)³ = -512.

How many zeros does 500³ have at the end?

500³ = (5 × 10²)³ = 5³ × 10⁶ = 125,000,000, so it has six trailing zeros.

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