Question

If \(2^{x}=3,\) what does \(4^{-x}\) equal?

Short answer

\(4^{-x} = \frac{1}{9}\)

Step-by-step solution

- Rewrite the base

First, notice that we can rewrite the number 4 as a power of 2. Since 4 is equal to \(2^2\), we can express \(4^{-x}\) as \((2^2)^{-x}\).

- Simplify the expression

Next, simplify \((2^2)^{-x}\) using the properties of exponents. This becomes \(2^{2 \times (-x)}\), which simplifies to \(2^{-2x}\).

- Substitute the value of x

From the problem, we know that \(2^x = 3\). We need to express \(2^{-2x}\) in terms of this value. Notice that \(2^{-2x} = (2^x)^{-2}\).

- Apply the known value

Using the known value \(2^x = 3\), replace \(2^x\) with 3 in the expression \((2^x)^{-2}\). This gives \(3^{-2}\).

- Calculate the final value

Finally, calculate \(3^{-2}\). This is the same as \(\frac{1}{3^2}\), which simplifies to \(\frac{1}{9}\). Thus, \(4^{-x} = \frac{1}{9}\).

Key concepts

Properties of Exponents

Understanding the properties of exponents is key to solving exponential equations like the one presented. Let's explore the main rules you need. The power of a power property states \[ (a^m)^n = a^{mn} \], where the exponents multiply. Another important property is negative exponents, such as \[ a^{-n} = \frac{1}{a^n} \]. These properties are essential for simplifying and transforming exponential expressions.
In this exercise, we start by rewriting 4 as a power of 2: \[ 4^{-x} = (2^2)^{-x} \]. Then, we simplify using the power of a power property: \[ (2^2)^{-x} = 2^{-2x} \]. Recognizing and applying the correct exponent rules helps accurately manipulate and simplify these kinds of algebraic expressions.

Substitution in Equations

Substitution is a powerful algebraic tool. It lets you replace complex parts of an equation with simpler, known values. In this problem, we're given that \[ 2^x = 3 \]. We can use this relationship to express other related components.
After simplifying to \[ 2^{-2x} \], we can substitute \[ 2^x \] with 3, the given value. Let’s use this substitution: \[ 2^{-2x} = (2^x)^{-2} \]. By plugging in 3 for \[ 2^x \], the expression changes to \[ (3)^{-2} \]. This simplifies our computation significantly since working with numbers is often easier than handling the variables.

Simplifying Algebraic Expressions

Once we’ve applied substitution, the next step is to simplify the resultant expression. When handling exponents, it’s vital to follow the proper algebraic steps to ensure accuracy.
Given our expression from substitution, \[ (3)^{-2} \], we use the negative exponent property: \[ a^{-n} = \frac{1}{a^n} \]. Thus, \[ (3)^{-2} \] becomes \[ \frac{1}{3^2} \]. Calculating this gives \[ \frac{1}{9} \].
This makes it clear that the value of \[ 4^{-x} = \frac{1}{9} \]. Breaking down each step methodically not only gets the correct answer but also builds a solid understanding of algebraic manipulation and simplification.