Question

Compute \(4^{2^{-1}}\). A. 2 B. 16 C. -16 D. -2

Short answer

2

Step-by-step solution

Understand the exponent

The exponent is written as a fraction: \(2^{-1}\). This means you need the reciprocal of 2.

Simplify the exponent

The negative exponent \(2^{-1}\) can be rewritten as \(\frac{1}{2}\). This means \(4^{2^{-1}} = 4^{\frac{1}{2}}\).

Recall the meaning of fractional exponents

A fractional exponent like \(\frac{1}{2}\) represents a square root. So \(4^{\frac{1}{2}}\) is the same as \(\sqrt{4}\).

Compute the square root

The square root of 4 is 2. Therefore, \(\sqrt{4} = 2\).

Verify the result

Verify that there are no other possible values or interpretations. The only correct answer is 2.

Key concepts

Reciprocal

When we talk about the 'reciprocal' of a number, we mean the number you multiply by to get 1. For example, the reciprocal of 2 is \(\frac{1}{2}\), because \(2 \times \frac{1}{2} = 1\). This is key when dealing with negative exponents. Remember that \(a^{-1}\) is the same as \(\frac{1}{a}\). So in our original exercise, \(2^{-1}\) becomes \(\frac{1}{2}\). This transformation helps simplify problems involving negative exponents. Simple, right? This is a tool you will use frequently in algebra and beyond.

Negative Exponents

Negative exponents can seem confusing at first, but they follow a simple rule. A negative exponent means you take the reciprocal of the base number. For instance, \(3^{-1} = \frac{1}{3}\). Therefore, \(2^{-1}\) converts into \(\frac{1}{2}\). This is a direct use of the reciprocal concept. For larger exponents, apply the negative exponent to the entire expression: \(a^{-n} = \frac{1}{a^n}\). So \(4^{2^{-1}}\) translates to \(4^{\frac{1}{2}}\). Think of the negative exponent sign as an instruction to flip the number fractionally. This might seem minor, but it's very powerful.

Square Root

Fractional exponents can be tricky, but they often represent roots. For example, \(4^{\frac{1}{2}}\) is the same as \(\text{sqrt}(4)\). The fractional exponent \(\frac{1}{2}\) means you are finding the square root of the base, in this case, 4. Knowing that 2 multiplied by itself equals 4 helps you see that \(\text{sqrt}(4) = 2\). This idea extends to other roots as well, like cube roots for \(\frac{1}{3}\) and so on. Understanding this can seriously simplify your calculations and helps you recognize patterns between exponents and roots. Once you internalize this, math involving roots and exponents becomes much more intuitive.