Problem 78 - Beginning and Intermediate Algebra

Question

Solve. $$t^{1 / 5}=2$$

Step-by-Step Solution

Answer

The solution to the given equation \(t^{\frac{1}{5}} = 2\) is \(t = 32\).

Step 1: Identify the equation given

The given equation is \(t^{\frac{1}{5}} = 2\).

Step 2: Raise both sides to the power of 5

To eliminate the exponent on the left side and isolate t, we will raise both sides to the power of 5. \[(t^{\frac{1}{5}})^5 = 2^5\]

Step 3: Simplify the equation

Apply the power rule (when raising a power to another power, multiply the exponents) to simplify the equation: \[t^{(1/5 * 5)} = 2^5\] This simplifies to: \[t^1 = 2^5\]

Step 4: Calculate the value of t

Now that we have simplified the equation, we can find the value of t by calculating the right-hand side of the equation: \[t = 2^5\] Calculating the value gives: \[t = 32\]

Step 5: Write the final answer

The solution to the given equation \(t^{\frac{1}{5}} = 2\) is \(t = 32\).

Key Concepts

Radicals

Radicals might sound like a complex topic, but they simply refer to expressions that involve roots, like square roots or cube roots. In our exercise, we encounter the fifth root, as indicated by the exponent of \( \frac{1}{5} \). This means we are looking for a number which, when raised to the fifth power, gives us the original number.
To clear a radical, you usually need to perform the reverse operation, which is raising the number to a power. For instance:

• The square root is undone by squaring the number.
• The cube root is undone by cubing it.
• For the fifth root, you raise it to the fifth power, as demonstrated in the exercise.

Radicals help us work with powers in simpler terms, allowing us to transform these expressions back into numbers we can more readily manipulate or use. Understanding radicals and their connection to exponents can demystify and simplify seemingly complicated algebraic expressions.

Power Rule

The power rule is an important principle when dealing with exponential expressions. It refers to a situation where you take an exponentiated value and raise it in turn to another power. The power rule states that you multiply the exponents together.
In mathematical terms, for any base \( a \) and exponents \( m \) and \( n \):

• \((a^m)^n = a^{m \cdot n}\)

In our example, the expression \((t^{\frac{1}{5}})^5\) illustrates this rule. By multiplying \( \frac{1}{5} \) and \( 5 \), we effectively undo the fifth root, simplifying it to \( t^1 \), which is just \( t \). This makes it much easier to solve for \( t \).
The power rule is fundamental for solving equations and simplifying expressions, playing a critical role in understanding how changes in exponents affect the terms involved. Its application is crucial in streamlining calculations and finding solutions efficiently.

Solving Equations

Solving equations involves finding the value of the unknown that makes the equation true. In algebra, this often requires isolating the variable on one side of the equal sign.
In the provided exercise, the goal was to find the value of \( t \). We started with an equation in the form of \( t^{\frac{1}{5}} = 2 \). The first step was to undo the fractional exponent, which we achieved by raising both sides to the fifth power. This isolated \( t \) on one side and transformed the problem into straightforward arithmetic by calculating \( 2^5 \), leading to:

• \( t = 32 \)

This procedure highlights the general approach to solving equations:

• Identify operations affecting the variable.
• Perform inverse operations to isolate the variable.
• Simplify the expression to find the solution.

Having a systematic approach to solving equations helps tackle both simple and complex problems by making clear, logical steps towards the desired solution.