Cube Root of 113

Last reviewed on 28 April 2026.

The cube root of 113 is approximately 4.8346 (more precisely, 4.83457032…). It is an irrational number, meaning its decimal expansion never terminates and never repeats. In symbols: ∛113 ≈ 4.8346, or 1131/3 ≈ 4.8346.

What "Cube Root of 113" Means

The cube root of a number x is the value that, when multiplied by itself three times, gives x. So "cube root of 113" asks: what number cubed equals 113?

It is the inverse of cubing. Just as 4³ = 64 and 5³ = 125, the answer for 113 has to fall somewhere between 4 and 5. A quick estimate: 4.8 cubed is 110.592; 4.9 cubed is 117.649. The true cube root sits closer to 4.83.

By contrast, the square root of 113 is about 10.6301 — a much larger value, because squaring grows more slowly than cubing in this range. Both roots are irrational because 113 is not a perfect square or a perfect cube.

Worked Calculation

Without a calculator, you can pin the cube root of 113 down by trial:

  1. 4.8³ = 4.8 × 4.8 × 4.8 = 23.04 × 4.8 = 110.592. Too small.
  2. 4.9³ = 4.9 × 4.9 × 4.9 = 24.01 × 4.9 = 117.649. Too large.
  3. Try 4.83: 4.83² = 23.3289, then 23.3289 × 4.83 = 112.679. Just under.
  4. Try 4.84: 4.84² = 23.4256, then 23.4256 × 4.84 = 113.380. Just over.
  5. 4.83 is closer to 113 than 4.84, so the cube root is 4.83 plus a small fraction.
  6. Linear interpolation: (113 − 112.679) / (113.380 − 112.679) ≈ 0.458, giving 4.83 + 0.01 × 0.458 ≈ 4.8346.

The same value falls out of any scientific calculator or programming language: 113 ** (1/3) in Python, or cbrt(113) in C, JavaScript, and most modern languages.

Why ∛113 Is Irrational

A number is a perfect cube only if its prime factorisation has every prime appearing to a power that is a multiple of three. The factorisation of 113 is just 113 itself — because 113 is a prime number. The prime 113 appears to the power one, not three.

That means there is no integer (and no fraction) whose cube is exactly 113. The cube root therefore cannot be written as p/q for integers p and q; in technical terms, it is irrational. Its decimal expansion runs forever without repeating.

Cube Root of 113 in Different Forms

  • Approximate decimal: 4.83457032465006…
  • To 4 decimal places: 4.8346
  • To 2 decimal places: 4.83
  • Exponent form: 1131/3
  • Surd form: ∛113 (cannot be simplified, because 113 has no cube factors other than 1)
  • Continued fraction (start): [4; 1, 4, 5, 1, 1, …]

Cube Root of 113 vs Square Root of 113

Putting the two roots side-by-side highlights a useful pattern:

  • √113 ≈ 10.6301 — the side length of a square with area 113.
  • ∛113 ≈ 4.8346 — the side length of a cube with volume 113.

Both are irrational; both come from 113's primality. But the cube root is much smaller because volume grows quickly with side length: a cube only needs sides of about 4.83 units to enclose 113 cubic units, whereas a square needs sides of 10.63 to enclose 113 square units.

You can verify the relationship: (∛113)³ = 113, while (√113)² = 113. They are different roots of the same number.

Where the Cube Root of 113 Shows Up

Cube roots appear whenever a problem connects volume, capacity, or three-dimensional scaling to a side length. A few practical places where ∛113 ≈ 4.8346 might come up:

  • Volume-to-side conversions. A cube of volume 113 cm³ has sides of about 4.83 cm. A cube of volume 113 m³ has sides of about 4.83 m.
  • Scaling laws. If a 3D object has its volume multiplied by 113, its linear dimensions multiply by ∛113 ≈ 4.83. The same logic links biological scaling, fluid mixing, and engineering models.
  • Standardised testing. Cube-root problems with non-perfect-cube inputs are common on quantitative reasoning sections; 113 is a clean example because students cannot reach for a memorised perfect cube.

Common Mistakes

  • Confusing ∛113 with √113. The cube root sits near 4.83; the square root sits near 10.63. The square root is more than twice as large.
  • Reading the calculator wrong. If you type 113^1/3 in a calculator that does not respect the order of operations correctly, you may get 113/3 ≈ 37.67. Use parentheses: 113^(1/3).
  • Trying to "simplify" the surd. ∛113 cannot be rewritten as a smaller surd, because 113 has no perfect-cube factor. Compare with ∛125 = 5, where the cube comes out cleanly.
  • Rounding too early. If you square or cube a rounded value of ∛113, you will not get exactly 113. Carry several decimals through your calculation, then round the final answer.

Quick-Reference Card

  • Cube root of 113: ≈ 4.8346
  • More precise: 4.83457032…
  • Verification: 4.8346³ ≈ 113.00 ✓
  • Type of number: irrational
  • Surd form: ∛113 (cannot be simplified)
  • Compare: √113 ≈ 10.6301

For more on 113's mathematical identity, see Is 113 prime?, factors of 113, and number 113 properties.