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Chapter 5: Problem 13

Find the value of \(t\) in each equation. \(t^{4}=81\)

Short Answer

Expert verified
t = \pm 3

Step by step solution

01

- Understand the Equation

Recognize that the equation is in the form of a polynomial equation. Here, the variable is raised to the fourth power: \(t^{4} = 81\).
02

- Take the Fourth Root of Both Sides

To solve for \(t\), take the fourth root of both sides of the equation: \[ t = \sqrt[4]{81} \]
03

- Simplify the Fourth Root

Calculate the fourth root of 81. Since \(81 = 3^{4}\), \[ \sqrt[4]{81} = 3 \]
04

- Consider Positive and Negative Roots

Since any number raised to an even power can have both positive and negative roots, \[t = \pm 3\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fourth Root
To solve the equation, we first need to understand the concept of the fourth root. The fourth root of a number is a value that, when raised to the power of four, gives the original number.
Mathematically, the fourth root of a number \(a\) is represented as \( \sqrt[4]{a} \). For the given problem, we need to find the fourth root of 81.
Breaking it down: if \( t^{4} = 81 \), then \( t = \sqrt[4]{81} \).
Knowing that \( 81 \) can be expressed as \( 3^{4} \), it simplifies the computation: \( \sqrt[4]{81} = 3 \).
Positive and Negative Roots
When dealing with polynomial equations, especially those raised to an even power, we must consider both positive and negative roots.
This is because any real number raised to an even power will give a non-negative result.
For instance, both \( 3^{4} \) and \( (-3)^{4} \) are equal to 81.
Therefore, the roots of the equation \( t^{4} = 81 \) are \( t = 3 \) and \( t = -3 \).
In summary: \
  • Positive root: \( t = 3 \)
  • Negative root: \( t = -3 \)

Hence, we write it as \( t = \pm 3 \).
Algebraic Simplification
Algebraic simplification is a critical step in solving polynomial equations. It involves reducing equations and expressions to their simplest forms.
In our example, the given equation is \( t^{4} = 81 \).
By taking the fourth root of both sides, we are simplifying the equation from \( t^{4} = 81 \) to \( t = \sqrt[4]{81} \).
We then further simplify this by recognizing \( 81 \) as \( 3^{4} \), leading to \( \sqrt[4]{81} = 3 \).
Simplification reduces the complexity of the problem, making it easier to find the solution.
Polynomial Equation
Understanding polynomial equations is essential for solving them effectively. A polynomial equation involves a variable raised to a non-negative integer exponent.
In this problem, the variable \( t \) is raised to the fourth power: \( t^{4} = 81 \). This is an example of a fourth-degree polynomial equation.
The general strategy to solve such equations involves:
  • Recognizing the polynomial form
  • Taking appropriate roots to simplify the equation
  • Considering all possible roots (positive and negative)

Mastering polynomial equations opens the door to solving more complex algebraic problems.

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Most popular questions from this chapter

Find the value of \(t\) in each equation. \(3^{t}=729\)

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