Problem 25 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(2^{2 x+1}=16^{x}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{1}{2}\).

Step 1: Rewrite 16 as a Power of 2

The equation given is \(2^{2x+1}=16^{x}\). First, rewrite 16 as a power of 2. We know that \(16 = 2^4\). So, we can rewrite the right side as \((2^4)^x\).

Step 2: Apply the Power of a Power Rule

Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we rewrite \((2^4)^x\) as \(2^{4x}\). The equation now becomes \(2^{2x+1} = 2^{4x}\).

Step 3: Set Exponents Equal

Since the bases are the same (both sides have a base of 2), we can set the exponents equal to each other: \[2x+1 = 4x\]

Step 4: Solve for x

Solve the equation \(2x + 1 = 4x\):Subtract \(2x\) from both sides:\[1 = 2x\]Divide both sides by 2:\[x = \frac{1}{2}\]

Step 5: Verify the Solution

Substitute \(x = \frac{1}{2}\) back into the original equation to verify:Left side: \(2^{2(\frac{1}{2})+1} = 2^2 = 4\)Right side: \(16^{\frac{1}{2}} = (2^4)^{\frac{1}{2}} = 2^{4 \cdot \frac{1}{2}} = 2^2 = 4\)Both sides equal, so \(x = \frac{1}{2}\) is verified to be correct.

Key Concepts

Laws of Exponents

The laws of exponents are a set of rules that make it easier to work with exponential expressions. They allow us to simplify and manipulate powers of numbers efficiently.
One key law is the **power of a power** rule, which states that

• \((a^m)^n = a^{m \cdot n}\).

This means when you have an exponent raised to another exponent, you multiply them together.
Another important law is the **same-base exponential equality**, which helps in solving equations like when both sides of the equation have the same base. We can safely say that the exponents themselves are equal:

• If \(a^m = a^n\), then \(m = n\).

These rules simplify complex algebraic expressions and are crucial when working with exponential equations as they allow us to focus on the exponents themselves rather than the base.

Algebraic Manipulation

Algebraic manipulation involves rewriting and rearranging equations and expressions to uncover insights or find solutions. In the context of exponential equations, algebraic manipulation often involves:

• Changing the form of numbers, as seen in the exercise above where \(16\) was rewritten as \(2^4\).
• Applying laws of exponents to consolidate or simplify exponential terms.

Rewriting equations and expressions in equivalent forms helps to reveal solutions by making relationships between variables or coefficients much clearer.
In the given example, simple algebraic manipulations helped to equate both sides to a single base, uncovering the core equality through restating the problem in simpler terms.Steps forward often involve isolating variables or terms of interest, which provides clarity and facilitates efficient problem-solving.

Powers of Numbers

Understanding powers of numbers enables a deeper grasp of exponential relations. A power consists of a base and an exponent, organized in the form \(a^b\), where \(a\) is the base, and \(b\) is the exponent.
In this scenario:

• Raising a number to a power is akin to multiplying that number by itself as many times as the exponent states.
• The base is the constant factor, while the exponent indicates the number of repetitions.

For the number 16, knowing that it can be expressed as \(2^4\) transforms the problem completely by aligning it under a unified base for both sides of the equation.
This makes complex equations like \(2^{2x+1}=16^x\) much simpler to tackle, as it gives a base for comparison. The ability to adeptly manage powers of numbers is key to efficiently solving exponential equations.