2 to the 1/4 Power?

2 to the 1/4 Power?

2 to the power of 1/4, written as 2^(1/4), is equal to the fourth root of 2. To calculate it, you find the number that, when raised to the fourth power, equals 2. It is approximately equal to 1.4142. So, 2^(1/4) is the same as the fourth root of 2, which is approximately 1.4142.

Simplifying Exponents with Fractions

Exponents with fractional bases require a different process than whole number exponents. When the exponent is a fraction, we can’t just repeatedly multiply the base. Instead, we rely on the definition of roots to evaluate fractional exponents.

Let’s look at simplifying 2^(1/4) using the meaning of the fractional exponent.

Understanding Roots

A root is the inverse operation of an exponent. For example:

Square root: √ = 1/2 exponent Cube root: 3√ = 1/3 exponent

So a fractional exponent takes a root to get rid of the denominator:

2^(1/4) = 4√2

This turns the expression into a more familiar radical form.

Simplifying the Root

To evaluate 4√2:

1. Factor out perfect squares: 4√2 = (4 * √2)
2. Take the square root of the perfect square: 4 * √2 = 4 * 1.414
3. Multiply: = 5.656

So 2^(1/4) = 5.656 when simplified.

Why This Works

A fractional exponent 1/n takes the nth root of the base:

2^(1/4) = 4√2

Because 4th root “undoes” squaring 4 times. This method helps make exponents with fractions more intuitive.

In summary, converting a fractional exponent to a radical simplifies evaluation. Understanding exponents as repeated multiplication allows us to reason through fractional powers by relating them to roots as the inverse operation.