Question

If \(4^{x}-4^{x-1}=24,\) what is the value of \(\left(2^{x}\right)^{x} ?\)

Short answer

The value of \((2^x)^x\) is \(2^{\frac{25}{4}}\).

Step-by-step solution

Simplify the given equation

The given equation is \(4^{x} - 4^{x-1} = 24\). Notice that \(4^{x}\) can be written as \((2^2)^{x} = 2^{2x}\) and \(4^{x-1}\) as \((2^{2})^{x-1} = 2^{2(x-1)}\).

Rewrite the equation with power of two

Now substitute \(2^{2x}\) for \(4^{x}\) and \(2^{2x-2}\) for \(4^{x-1}\) into the equation: \(2^{2x} - 2^{2(x-1)} = 24\).

Simplify the expression

Simplify \(2^{2x-2}\) on the left-hand side of the equation: \(2^{2x-2} = \frac{2^{2x}}{2^{2}} = \frac{2^{2x}}{4}\). Therefore, the equation becomes: \(2^{2x} - \frac{2^{2x}}{4} = 24\).

Factor out common terms

Factor \(2^{2x}\) out from the left-hand side: \(2^{2x}\left(1 - \frac{1}{4}\right) = 24\).

Simplify the equation

Simplify the term inside the parentheses: \(2^{2x}\left(\frac{3}{4}\right) = 24\).

Solve for \(2^{2x}\)

Multiply both sides by \(\frac{4}{3}\) to isolate \(2^{2x}\): \(2^{2x} = 24 \cdot \frac{4}{3} = 32\).

Find \(x\)

Since \(2^{2x} = 32\) and \(32 = 2^5\), equate the exponents: \(2x = 5\). Thus, \(x = \frac{5}{2}\).

Calculate \((2^x)^x\)

Finally, to find the value of \((2^x)^x\), substitute \(x=\frac{5}{2}\) into the expression: \((2^{\frac{5}{2}})^{\frac{5}{2}} = 2^{\frac{5}{2} \cdot \frac{5}{2}} = 2^{\frac{25}{4}}\).

Key concepts

Power of Two

Understanding how to work with powers of two is essential in solving exponential equations. In our given problem, the bases are initially in the form of 4, but we recognize that 4 can be rewritten as a power of 2. Specifically, \(4 = 2^2\). This transformation simplifies our calculations significantly.
For example, converting \(4^x\) to \(2^{2x}\) involves recognizing the exponent multiplication rule: \( (a^m)^n = a^{m \times n} \). By rewriting all terms with a consistent base, we can streamline the equations and simplify further steps.

Simplification

Simplification is the process of rewriting mathematical expressions in a more manageable form. In our exercise, we start by converting \(4^x\) and \(4^{x-1}\) into powers of two. This gives us \(2^{2x}\) and \(2^{2(x-1)}\). The next step involves simplifying the expression \(2^{2(x-1)}\):
\(2^{2(x-1)} = 2^{2x-2}\).
Here, we apply the property \( a^{m-n} = \frac{a^m}{a^n} \). This tells us \(2^{2x-2} = \frac{2^{2x}}{2^2} = \frac{2^{2x}}{4}\), simplifying our equation to a form where we can easily combine like terms.

Exponent Rules

Exponent rules are crucial for manipulating expressions involving powers. In this problem, we apply several key exponent rules:

• Rule of Rewriting Powers: \(a^m * a^n = a^{m+n}\).
• Rule of Fractional Exponents: \(a^{m-n} = \frac{a^m}{a^n}\).

Using these rules, we factor out \(2^{2x}\) from the equation: \(2^{2x} - \frac{2^{2x}}{4} = 24\), and then simplify the factor: \(2^{2x} (1 - \frac{1}{4}) = 24\), leading to \(2^{2x} * \frac{3}{4} = 24\). With these rules, we solved for \(2^{2x}\) cleanly and efficiently.

Factoring

Factoring is the process of breaking down expressions into simpler components that can be multiplied together. In our problem, we've simplified our main equation to \(2^{2x} * \frac{3}{4} = 24\). To isolate \(2^{2x}\), we factor out and simplify the components:

• First, multiply both sides by the reciprocal of \(\frac{3}{4}\): \(\frac{4}{3} * 24 = 32\).
• Next, recognize that \(32\) is a power of 2: \(32 = 2^5\).

Equating \(2^{2x} = 2^5\) allows us to solve for x: \(2x = 5 \Rightarrow x = \frac{5}{2}\). Finally, substituting this solution back into our expression \((2^x)^x\), we find \( (2^{5/2})^{5/2} = 2^{25/4}\).