Problem 14 - Algebra 2

Question

Solve each equation. Check your solution. $$ 4^{3 x+2}=\frac{1}{256} $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Express both sides with the same base

To solve the equation \( 4^{3x+2} = \frac{1}{256} \), express both sides as powers of 4. Since 256 is a power of 4, rewrite \( \frac{1}{256} \) as \( 4^{-4} \), because \( 4^4 = 256 \). Thus, the equation is rewritten as \( 4^{3x+2} = 4^{-4} \).

Step 2: Set exponents equal to each other

Now that both sides of the equation have the same base, set the exponents equal to each other: \[ 3x + 2 = -4 \]

Step 3: Solve for x

Solve the equation \( 3x + 2 = -4 \) for \( x \). Start by subtracting 2 from both sides:\[ 3x = -4 - 2 \] \[ 3x = -6 \]Then divide both sides by 3:\[ x = \frac{-6}{3} \] Thus, \( x = -2 \).

Step 4: Check the solution

To verify the solution, substitute \( x = -2 \) back into the original equation:\[ 4^{3(-2)+2} = \frac{1}{256} \]Calculate the exponent:\[ 4^{-6 + 2} = 4^{-4} \]So on both sides we have \( 4^{-4} = \frac{1}{256} \), which confirms the solution is correct.

Key Concepts

Powers of Numbers

Understanding powers or exponents is an essential part of solving exponent equations. A power essentially represents multiplication of a number by itself a certain number of times. For instance, the expression \( 4^3 \) means \( 4 \times 4 \times 4 \). Here, 4 is the base, and 3 is the exponent. Exponents serve to simplify repeated multiplication and make larger computations manageable.

• The number being multiplied, such as 4 in our examples, is called the base.
• The number indicating how many times the base is multiplied is the exponent.
• The expression as a whole is called a power.

Get comfortable recognizing common powers of numbers, like \( 4^2 = 16 \) and \( 4^4 = 256 \). This helps in recognizing number patterns needed for equation solving. Remember that any number raised to the power of zero equals 1, and a negative exponent means taking the reciprocal of the base raised to the positive exponent form.

Equation Solving

Equation solving is the process of finding the value for unknown variables that make an equation true. In exponent equations, this involves multiple steps of manipulation to isolate the unknown variable. Here’s a simplified view of how we solve an equation like \( 4^{3x+2} = \frac{1}{256} \):

• First, express each side of the equation using powers of the same base, if possible.
• Next, because the bases are the same, their exponents must also be equal, hence equate the exponents.
• Finally, solve the resulting algebraic equation for the variable.

For effective equation solving, it's vital to follow these steps systematically, ensuring each transformation maintains equality, leading to the correct solution.

Base and Exponent

Understanding the distinction between a base and an exponent is crucial in manipulating exponent equations. In expressions like \( 4^{3x+2} \), 4 is the base, and \( 3x+2 \) is the exponent. The base is the number being repeatedly multiplied, while the exponent indicates how many times the base is used as a factor.

• Converting numbers to the same base allows for simplifying equations. For example, expressing \( \frac{1}{256} \) as \( 4^{-4} \) because \( 256 = 4^4 \).
• Exponents can be simplified or transformed through addition, subtraction, multiplication, or other algebraic methods once they are isolated.

Grasping the concept of bases and exponents empowers us to rewrite complex numbers more simply and solve equations efficiently. This skill helps keep both sides of an equation balanced during manipulation.

Checking Solutions

Once you solve an equation, checking your solution is a critical step to confirm its accuracy. In exponent equations, this involves substituting the value of the variable back into the original equation. Let's apply this to check our solution \( x = -2 \):

• Substitute \( x = -2 \) into the original equation: \( 4^{3(-2)+2} \).
• Simplify: \( 4^{-6+2} = 4^{-4} \).
• Confirm the simplified value matches the right side of the equation: \( \frac{1}{256} = 4^{-4} \).

This process ensures the calculated solution truly satisfies the equation, proving it correct. Always perform this check to avoid common errors and ensure understanding of exponent manipulation.