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Simplifying Fourth Roots Calculator

Reviewed by Calculator Editorial Team

A fourth root of a number is a value that, when raised to the power of 4, gives the original number. Simplifying fourth roots involves expressing them in their simplest radical form or as exact values when possible.

What is a Fourth Root?

The fourth root of a number \( x \) is a number \( y \) such that \( y^4 = x \). For example, the fourth root of 16 is 2 because \( 2^4 = 16 \).

Fourth roots can be expressed in radical form as \( \sqrt[4]{x} \). When \( x \) is a perfect fourth power, the fourth root is an integer. Otherwise, the radical form is simplified as much as possible.

How to Simplify Fourth Roots

To simplify a fourth root, follow these steps:

  1. Factor the radicand (the number under the radical) into perfect fourth powers and other factors.
  2. Separate the radical into the product of the fourth roots of each factor.
  3. Take the fourth root of any perfect fourth powers.
  4. Combine the results to get the simplified form.

Formula: \( \sqrt[4]{a \times b} = \sqrt[4]{a} \times \sqrt[4]{b} \)

For example, to simplify \( \sqrt[4]{32} \):

  1. Factor 32 into \( 16 \times 2 \), where 16 is a perfect fourth power.
  2. Separate the radical: \( \sqrt[4]{16 \times 2} = \sqrt[4]{16} \times \sqrt[4]{2} \).
  3. Take the fourth root of 16: \( 2 \times \sqrt[4]{2} \).

The simplified form is \( 2\sqrt[4]{2} \).

Worked Examples

Example 1: Simplifying \( \sqrt[4]{81} \)

81 is a perfect square (9 × 9), but not a perfect fourth power. However, we can express it as \( 9^2 \), which is \( (3^2)^2 = 3^4 \).

Therefore, \( \sqrt[4]{81} = \sqrt[4]{3^4} = 3 \).

Example 2: Simplifying \( \sqrt[4]{160} \)

Factor 160 into \( 16 \times 10 \), where 16 is a perfect fourth power.

Separate the radical: \( \sqrt[4]{16 \times 10} = \sqrt[4]{16} \times \sqrt[4]{10} \).

Take the fourth root of 16: \( 2 \times \sqrt[4]{10} \).

The simplified form is \( 2\sqrt[4]{10} \).

FAQ

What is the difference between a square root and a fourth root?

A square root is a number that, when multiplied by itself, gives the original number. A fourth root is a number that, when raised to the power of 4, gives the original number. Fourth roots are less common than square roots but follow similar simplification rules.

Can all fourth roots be simplified?

No, not all fourth roots can be simplified. Only fourth roots of perfect fourth powers (like 16, 81, 256) can be simplified to exact integer values. Other fourth roots remain in radical form.

How do I calculate a fourth root without a calculator?

For perfect fourth powers, you can recognize them as squares of perfect squares. For example, \( 2^4 = 16 \), so \( \sqrt[4]{16} = 2 \). For other numbers, you can use estimation or the Newton-Raphson method for approximation.