2.1 Solve For X X X : $\sqrt[x] 32} = 128$2.2 Solve The Equation $x^{\frac{1 {2}} - 3x^{\frac{1}{4}} - 10 = 0$

Introduction

In mathematics, equations involving exponents and roots can be challenging to solve. These types of equations often require a deep understanding of algebraic concepts and techniques. In this article, we will focus on solving two specific equations: $\sqrt[x]{32} = 128$ and $x^{\frac{1}{2}} - 3x^{\frac{1}{4}} - 10 = 0$. We will use various mathematical techniques to solve these equations and provide step-by-step solutions.

Solving the Equation $\sqrt[x]{32} = 128$

The given equation is $\sqrt[x]{32} = 128$. To solve this equation, we need to isolate the variable $x$. We can start by raising both sides of the equation to the power of $x$.

Step 1: Raise both sides to the power of $x$

$\left(\sqrt[x]{32}\right)^x = 128^x$

This simplifies to:

$32 = 128^x$

Step 2: Express $128$ as a power of $2$

$128 = 2^7$

Substituting this into the equation, we get:

$32 = (2^7)^x$

Step 3: Simplify the equation using exponent rules

Using the rule $(a^m)^n = a^{mn}$, we can simplify the equation:

$32 = 2^{7x}$

Step 4: Express $32$ as a power of $2$

$32 = 2^5$

Substituting this into the equation, we get:

$2^5 = 2^{7x}$

Step 5: Equate the exponents

Since the bases are the same, we can equate the exponents:

$5 = 7x$

Step 6: Solve for $x$

Dividing both sides by $7$, we get:

$x = \frac{5}{7}$

Therefore, the solution to the equation $\sqrt[x]{32} = 128$ is $x = \frac{5}{7}$.

Solving the Equation $x^{\frac{1}{2}} - 3x^{\frac{1}{4}} - 10 = 0$

The given equation is $x^{\frac{1}{2}} - 3x^{\frac{1}{4}} - 10 = 0$. To solve this equation, we need to isolate the variable $x$. We can start by factoring out a common term.

Step 1: Factor out a common term

$x^{\frac{1}{4}}\left(x^{\frac{1}{4}} - 3\right) - 10 = 0$

Step 2: Add $10$ to both sides

$x^{\frac{1}{4}}\left(x^{\frac{1}{4}} - 3\right) = 10$

Step 3: Factor the left-hand side

$\left(x^{\frac{1}{4}}\right)^2 - 3x^{\frac{1}{4}} - 10 = 0$

Step 4: Let $y = x^{\frac{1}{4}}$

Substituting $y$ into the equation, we get:

$y^2 - 3y - 10 = 0$

Step 5: Factor the quadratic equation

$(y - 5)(y + 2) = 0$

Step 6: Solve for $y$

Setting each factor equal to $0$, we get:

$y - 5 = 0$ or $y + 2 = 0$

Solving for $y$, we get:

$y = 5$ or $y = -2$

Step 7: Substitute back $y = x^{\frac{1}{4}}$

Substituting $y = x^{\frac{1}{4}}$ back into the equation, we get:

$x^{\frac{1}{4}} = 5$ or $x^{\frac{1}{4}} = -2$

Step 8: Raise both sides to the power of $4$

$x = 5^4$ or $x = (-2)^4$

Step 9: Simplify the equation

$x = 625$ or $x = 16$

Therefore, the solutions to the equation $x^{\frac{1}{2}} - 3x^{\frac{1}{4}} - 10 = 0$ are $x = 625$ and $x = 16$.

Conclusion

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Q: What is the difference between a root and an exponent?

A: A root and an exponent are related but distinct concepts in mathematics. An exponent is a number that is raised to a power, while a root is the inverse operation of an exponent. For example, $2^3$ is an exponent, while $\sqrt[3]{8}$ is a root.

Q: How do I solve an equation with a root?

A: To solve an equation with a root, you need to isolate the variable. This can involve raising both sides of the equation to a power, expressing numbers as powers of a base, equating exponents, and factoring quadratic equations.

Q: What is the order of operations for solving equations with exponents and roots?

A: The order of operations for solving equations with exponents and roots is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Roots: Evaluate any radical expressions (roots) next.
4. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
5. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the rule $(a^m)^n = a^{mn}$. This rule allows you to combine multiple exponents into a single exponent.

Q: What is the difference between a rational exponent and an irrational exponent?

A: A rational exponent is an exponent that can be expressed as a fraction, while an irrational exponent is an exponent that cannot be expressed as a fraction. For example, $2^{\frac{1}{2}}$ is a rational exponent, while $\sqrt[3]{8}$ is an irrational exponent.

Q: How do I solve an equation with a rational exponent?

A: To solve an equation with a rational exponent, you can use the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. This rule allows you to rewrite a rational exponent as a root.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, $x + 2 = 3$ is a linear equation, while $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula allows you to find the solutions to a quadratic equation.

Q: What is the difference between a real solution and a complex solution?

A: A real solution is a solution that can be expressed as a real number, while a complex solution is a solution that cannot be expressed as a real number. For example, $x = 2$ is a real solution, while $x = 2 + 3i$ is a complex solution.

Q: How do I determine whether a solution is real or complex?

A: To determine whether a solution is real or complex, you can use the discriminant of the quadratic equation. If the discriminant is positive, the solution is real. If the discriminant is negative, the solution is complex.

Conclusion

In this article, we answered some frequently asked questions about solving equations with exponents and roots. We covered topics such as the difference between a root and an exponent, the order of operations for solving equations with exponents and roots, and how to simplify expressions with multiple exponents. We also discussed the difference between a rational exponent and an irrational exponent, and how to solve equations with rational exponents. Finally, we covered the difference between a linear equation and a quadratic equation, and how to solve quadratic equations using the quadratic formula.