Problem 79 - Beginning and Intermediate Algebra

Question

Solve. $$t^{1 / 4}=3$$

Step-by-Step Solution

Answer

The solution to the equation \(t^{\frac{1}{4}} = 3\) is \(t = 81\).

Step 1: Write down the given equation

We are given the equation: \(t^{\frac{1}{4}} = 3\)

Step 2: Raise both sides to the power of 4

To eliminate the 1/4th power on variable t, we will perform an operation that is the inverse of taking the 1/4th power, which is raising to the power of 4. Take both sides and raise them to the power of 4: \((t^{\frac{1}{4}})^4 = 3^4\)

Step 3: Simplify the equation

After raising both sides of the equation to the power of 4, we can simplify the equation: \(t = 81\) The solution to the equation \(t^{\frac{1}{4}} = 3\) is \(t = 81\).

Key Concepts

Solving Equations

When you're confronted with an equation, like the one we have in our original exercise, solving it means finding the value of the variable that makes the equation true. In our case, the variable is \( t \), and the equation is \( t^{\frac{1}{4}} = 3 \).

To solve the equation, you'll need to perform operations that isolate the variable on one side of the equation; this helps you determine its value. These operations include adding, subtracting, multiplying, dividing, and even applying powers or roots. Each step taken during the process should maintain the balance of the equation: whatever you do to one side, you must also do to the other.

Our goal is to keep manipulating the equation until \( t \) is by itself on one side, and when it is, we know what \( t \) equals. For our example, by applying the next two key concepts—inverse operations and algebraic manipulation—we incrementally isolate \( t \) to solve for it correctly.

Inverse Operations

Inverse operations are like undoing what has been done. If you have an operation applied to a variable, the inverse operation will help reverse that effect.

Consider addition and subtraction; they are inverse operations of each other. Similarly, multiplication and division are inverse operations. In our given equation \( t^{\frac{1}{4}} = 3 \), the variable \( t \) is raised to the power of \( \frac{1}{4} \). The inverse operation here would be raising it to the power of 4.

This is because raising and taking a root are inverse operations. Raising \( t^{\frac{1}{4}} \) to the fourth power (\((t^{\frac{1}{4}})^4\)) essentially cancels out the \( \frac{1}{4} \) exponent, thereby isolating \( t \). Performing the same operation on the other side of the equation (\( 3^4 \)) keeps the equation balanced, resulting in \( t = 81 \).

Understanding inverse operations is crucial as it allows us to manipulate equations and find solutions more effectively.

Algebraic Manipulation

Algebraic manipulation is the heart of solving equations as it involves altering equations to simplify them or to isolate the variable. In our example, we started with \( t^{\frac{1}{4}} = 3 \).

To manipulate this equation, we applied the inverse operation found in the previous section. Applying the fourth power to both sides simplifies the left-hand side, removing the fractional exponent from \( t \). The right side of the equation becomes \( 81 \) because \( 3^4 = 81 \).

Through algebraic manipulation, which involved careful selection of operations, we simplified the equation from its original complex state to a simple statement of \( t = 81 \).

Mastering algebraic manipulation is essential for solving more complex equations. It often involves thoughtful addition of inverse operations, thus leading to the solution while maintaining balance throughout the process.