Evaluate each exponential expression: 2^(-4) + 4^(-1)

Verified step by step guidance

1. Rewrite each term with a negative exponent using the property of exponents: a^(-n) = 1/(a^n). For the first term, rewrite 2^(-4) as 1/(2^4). For the second term, rewrite 4^(-1) as 1/(4^1).
2. Simplify the denominators of each fraction. For 1/(2^4), calculate 2^4, which means multiplying 2 by itself 4 times. For 1/(4^1), calculate 4^1, which is simply 4.
3. Substitute the simplified values of the denominators back into the fractions. This will give you two fractions to add together.
4. Find a common denominator for the two fractions. The denominators are powers of 2, so determine the least common multiple (LCM) of the denominators.
5. Rewrite each fraction with the common denominator, then add the numerators together. Simplify the resulting fraction if possible.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of a^x, where 'a' is a positive constant and 'x' is a variable exponent. These functions exhibit rapid growth or decay depending on the value of 'x'. Understanding how to evaluate these functions, especially with negative exponents, is crucial for solving problems involving exponential expressions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, a^(-n) is equivalent to 1/(a^n). This concept is essential for simplifying expressions with negative exponents, allowing for easier calculations and evaluations of exponential expressions.

Addition of Exponential Terms

When evaluating expressions that involve the addition of exponential terms, it is important to first simplify each term individually before combining them. This involves calculating the value of each exponential expression and then performing the addition. Understanding how to handle these operations is key to accurately solving the given expression.