Solve The Equation: ( 4 X 4 ) 2 = 1 \left(4 X^4\right)^2=1 ( 4 X 4 ) 2 = 1

$$

Introduction

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to approach and solve various types of equations. In this article, we will focus on solving the equation $\left(4 x^4\right)^2=1$. This equation involves exponentiation and can be solved using various mathematical techniques.

Understanding the Equation

The given equation is $\left(4 x^4\right)^2=1$. To solve this equation, we need to understand the properties of exponents and how to manipulate them. The equation involves a squared term, which means we need to find the value of $x$ that satisfies the equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by expanding the squared term using the power rule of exponents. The power rule states that $(a^m)^n = a^{mn}$. Applying this rule to the given equation, we get:

$$\left(4 x^4\right)^2 = 4^2 \cdot (x^4)^2 = 16 \cdot x^8$$

Step 2: Set Up the Equation

Now that we have simplified the equation, we can set it up as an equation:

$$16 \cdot x^8 = 1$$

Step 3: Solve for $x$

To solve for $x$, we can start by dividing both sides of the equation by 16:

$$x^8 = \frac{1}{16}$$

Step 4: Take the Eighth Root

To find the value of $x$, we can take the eighth root of both sides of the equation:

$$x = \sqrt[8]{\frac{1}{16}}$$

Step 5: Simplify the Expression

To simplify the expression, we can rewrite $\frac{1}{16}$ as $\frac{1}{2^4}$:

$$x = \sqrt[8]{\frac{1}{2^4}}$$

Step 6: Apply the Power Rule

Using the power rule, we can rewrite the expression as:

$$x = \frac{1}{2^{\frac{4}{8}}}$$

Step 7: Simplify the Fraction

To simplify the fraction, we can rewrite $\frac{4}{8}$ as $\frac{1}{2}$:

$$x = \frac{1}{2^{\frac{1}{2}}}$$

Step 8: Evaluate the Expression

To evaluate the expression, we can rewrite $2^{\frac{1}{2}}$ as $\sqrt{2}$:

$$x = \frac{1}{\sqrt{2}}$$

Step 9: Rationalize the Denominator

To rationalize the denominator, we can multiply both the numerator and denominator by $\sqrt{2}$:

$$x = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

Conclusion

In this article, we solved the equation $\left(4 x^4\right)^2=1$ using various mathematical techniques. We started by simplifying the equation, setting it up as an equation, solving for $x$, taking the eighth root, simplifying the expression, applying the power rule, simplifying the fraction, evaluating the expression, and rationalizing the denominator. The final solution is $x = \frac{\sqrt{2}}{2}$.

Additional Tips and Tricks

• When solving equations involving exponents, it is essential to understand the properties of exponents and how to manipulate them.
• When simplifying expressions, it is crucial to apply the power rule and simplify fractions.
• When evaluating expressions, it is essential to rationalize the denominator to ensure that the expression is in its simplest form.

Real-World Applications

Solving equations involving exponents has numerous real-world applications. For example, in physics, the equation of motion for an object under constant acceleration is given by $s = ut + \frac{1}{2}at^2$, where $s$ is the displacement, $u$ is the initial velocity, $t$ is the time, and $a$ is the acceleration. This equation involves exponentiation and can be solved using various mathematical techniques.

Final Thoughts

Solving equations involving exponents is a fundamental concept in mathematics, and it is essential to understand how to approach and solve various types of equations. By following the steps outlined in this article, you can solve equations involving exponents and apply the techniques to real-world problems.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Further Reading

• [1] "Solving Equations Involving Exponents" by Khan Academy
• [2] "Exponents and Logarithms" by MIT OpenCourseWare
• [3] "Mathematics for Computer Science" by Stanford University

Related Topics

• [1] "Solving Quadratic Equations"
• [2] "Solving Linear Equations"
• [3] "Solving Systems of Equations"

Tags

• [1] "Mathematics"
• [2] "Algebra"
• [3] "Calculus"
• [4] "Exponents"
• [5] "Equations"
• [6] "Solving Equations"
• [7] "Real-World Applications"
• [8] "Mathematical Techniques"
  

Introduction

Solving equations involving exponents can be a challenging task, but with the right techniques and strategies, it can be made easier. In this article, we will answer some of the most frequently asked questions about solving equations involving exponents.

Q1: What is the first step in solving an equation involving exponents?

A1: The first step in solving an equation involving exponents is to simplify the equation by applying the power rule of exponents. This rule states that $(a^m)^n = a^{mn}$.

Q2: How do I simplify an expression involving exponents?

A2: To simplify an expression involving exponents, you can apply the power rule of exponents. For example, if you have the expression $(2^3)^4$, you can simplify it by applying the power rule: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q3: What is the difference between a positive exponent and a negative exponent?

A3: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal. For example, $2^3$ means $2$ raised to the power of $3$, while $2^{-3}$ means $2$ raised to the power of $-3$ and then taken as a reciprocal.

Q4: How do I solve an equation involving a negative exponent?

A4: To solve an equation involving a negative exponent, you can start by isolating the negative exponent on one side of the equation. Then, you can apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$.

Q5: What is the rule for multiplying exponents?

A5: The rule for multiplying exponents states that when you multiply two numbers with the same base, you add their exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

Q6: How do I solve an equation involving a product of exponents?

A6: To solve an equation involving a product of exponents, you can start by applying the rule for multiplying exponents. Then, you can simplify the expression and solve for the variable.

Q7: What is the rule for dividing exponents?

A7: The rule for dividing exponents states that when you divide two numbers with the same base, you subtract their exponents. For example, $\frac{2^3}{2^4} = 2^{3-4} = 2^{-1}$.

Q8: How do I solve an equation involving a quotient of exponents?

A8: To solve an equation involving a quotient of exponents, you can start by applying the rule for dividing exponents. Then, you can simplify the expression and solve for the variable.

Q9: What is the rule for raising a power to a power?

A9: The rule for raising a power to a power states that when you raise a power to a power, you multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q10: How do I solve an equation involving a power raised to a power?

A10: To solve an equation involving a power raised to a power, you can start by applying the rule for raising a power to a power. Then, you can simplify the expression and solve for the variable.

Conclusion

Solving equations involving exponents can be a challenging task, but with the right techniques and strategies, it can be made easier. By understanding the rules for exponents and applying them correctly, you can solve equations involving exponents and apply the techniques to real-world problems.

Additional Tips and Tricks

• When solving equations involving exponents, it is essential to understand the properties of exponents and how to manipulate them.
• When simplifying expressions, it is crucial to apply the power rule and simplify fractions.
• When evaluating expressions, it is essential to rationalize the denominator to ensure that the expression is in its simplest form.

Real-World Applications

Solving equations involving exponents has numerous real-world applications. For example, in physics, the equation of motion for an object under constant acceleration is given by $s = ut + \frac{1}{2}at^2$, where $s$ is the displacement, $u$ is the initial velocity, $t$ is the time, and $a$ is the acceleration. This equation involves exponentiation and can be solved using various mathematical techniques.

Final Thoughts

Solving equations involving exponents is a fundamental concept in mathematics, and it is essential to understand how to approach and solve various types of equations. By following the steps outlined in this article, you can solve equations involving exponents and apply the techniques to real-world problems.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Further Reading

• [1] "Solving Equations Involving Exponents" by Khan Academy
• [2] "Exponents and Logarithms" by MIT OpenCourseWare
• [3] "Mathematics for Computer Science" by Stanford University

Related Topics

• [1] "Solving Quadratic Equations"
• [2] "Solving Linear Equations"
• [3] "Solving Systems of Equations"

Tags

• [1] "Mathematics"
• [2] "Algebra"
• [3] "Calculus"
• [4] "Exponents"
• [5] "Equations"
• [6] "Solving Equations"
• [7] "Real-World Applications"
• [8] "Mathematical Techniques"