Calculate fractional indices by finding the missing base, index, or value in V = a^n, or evaluate a base raised to p/q with common root options.
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Fractional Index Formula
The following formula is used to calculate the value of a number with a fractional index (rational exponent).
V = a^{\frac{p}{q}}Variables:
- V is the value of the expression
- a is the base number
- p is the numerator of the fractional exponent (the power)
- q is the denominator of the fractional exponent (the root), with q ≠ 0
To calculate a^(p/q), you can raise the base number to the power p/q directly, or (when working with real numbers) take the qth root of a and then raise the result to the power p. The special case p = 1 gives the nth-root form V = a^(1/q).
What is a Fractional Index?
A fractional index, also known as a rational exponent, is an exponent that is a fraction. It is used to represent both roots and powers of a number. The numerator of the fraction indicates the power to which the number is raised, while the denominator indicates the root of the number. For example, in the expression 2^(3/2), the number 2 is raised to the power of 3 and then the square root is taken (equivalently, for positive bases, you can take the square root first and then cube).
How to Calculate Fractional Index?
The following steps outline how to calculate the Fractional Index using the formula: V = a^(p/q).
- First, determine the base number (a).
- Next, determine the numerator (p) and denominator (q) of the fractional exponent.
- Next, substitute the values of a, p, and q into the formula V = a^(p/q).
- Finally, calculate the value of V (for example, take the qth root of a and then raise it to p).
- After inserting the variables and calculating the result, check your answer with a calculator.
Example Problem:
Use the following variables as an example problem to test your knowledge.
Base number (a) = 4
Fractional index (p/q) = 1/2 (p = 1, q = 2), so V = 4^(1/2) = 2
