How to Solve a Number with a Fractional Power?

To solve a number raised to a fractional power, you'll convert the fractional exponent into a radical expression, making it easier to calculate. Here's how:

Understanding Fractional Exponents

A fractional exponent takes the form a^m/n, where:

• a is the base.
• m is the numerator (the power to which the base is raised).
• n is the denominator (the index of the root).

Converting to Radical Form

The fractional exponent a^m/n can be rewritten in radical form as follows:

• a^m/n = ⁿ√(a^m) or (ⁿ√a)^m

This means you can either:

1. Raise the base a to the power of m and then take the nth root of the result.
2. Take the nth root of the base a and then raise the result to the power of m.

The second approach is often simpler, especially when dealing with larger numbers, as it can result in smaller intermediate values.

Steps to Solve

1. Rewrite the expression: Convert the fractional exponent into its equivalent radical form.
2. Calculate the root: Find the nth root of the base.
3. Raise to the power: Raise the result from step 2 to the power of m.

Examples

Example 1: 4^3/2

1. Rewrite: 4^3/2 = (√4)³
2. Calculate the root: √4 = 2
3. Raise to the power: 2³ = 8

Therefore, 4^3/2 = 8

Example 2: 27^2/3

1. Rewrite: 27^2/3 = (³√27)²
2. Calculate the root: ³√27 = 3 (because 3 3 3 = 27)
3. Raise to the power: 3² = 9

Therefore, 27^2/3 = 9

*Example 3: x^1/2 x^1/3**

When multiplying exponential terms with the same base, add the exponents.

1. Add the fractions: 1/2 + 1/3 = 3/6 + 2/6 = 5/6
2. Combine: x^1/2 * x^1/3 = x^5/6

Key Considerations

• Negative Bases: Be mindful of negative bases, especially when the denominator of the fraction is even. For instance, (-4)^1/2 involves taking the square root of a negative number, resulting in a complex number.
• Simplifying Fractions: Always simplify the fractional exponent before converting to radical form. This will make the calculations easier.

Solving a number with a fractional power involves understanding the relationship between exponents and radicals, converting the expression into a manageable form, and then performing the necessary calculations.