Select The Correct Answer.What Is The Solution To The Equation? $\sqrt[5]{x+7}=-2$A. -39 B. -17 C. 25 D. No Solution
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Introduction
Radical equations are a type of algebraic equation that involves a variable under a radical sign. In this article, we will focus on solving radical equations with a specific type of radical, the fifth root. We will use the equation as an example to demonstrate the steps involved in solving radical equations.
Understanding Radical Equations
A radical equation is an equation that contains a variable under a radical sign. The radical sign is denoted by a small number, known as the index, which indicates the type of radical. In this case, we are dealing with a fifth root, which means the index is 5. The general form of a radical equation is:
where is the index, is the variable, is a constant, and is a number.
Solving the Equation
To solve the equation , we need to isolate the variable . The first step is to eliminate the radical sign by raising both sides of the equation to the power of 5.
This simplifies to:
Solving for x
Now that we have eliminated the radical sign, we can solve for by subtracting 7 from both sides of the equation.
Conclusion
In this article, we have demonstrated the steps involved in solving a radical equation with a fifth root. We started by understanding the general form of a radical equation and then applied the steps to solve the specific equation . The solution to the equation is .
Answer Options
Based on the solution we obtained, we can now evaluate the answer options.
- A. -39: This is the correct answer.
- B. -17: This is not the correct answer.
- C. 25: This is not the correct answer.
- D. No solution: This is not the correct answer.
The final answer is .
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Q: What is a radical equation?
A: A radical equation is an equation that contains a variable under a radical sign. The radical sign is denoted by a small number, known as the index, which indicates the type of radical.
Q: How do I solve a radical equation?
A: To solve a radical equation, you need to isolate the variable by eliminating the radical sign. This can be done by raising both sides of the equation to the power of the index.
Q: What is the index in a radical equation?
A: The index is the small number that appears in the radical sign. It indicates the type of radical and is used to determine the power to which both sides of the equation should be raised.
Q: How do I determine the power to which both sides of the equation should be raised?
A: The power to which both sides of the equation should be raised is equal to the index of the radical.
Q: What if the equation has a negative number under the radical sign?
A: If the equation has a negative number under the radical sign, you will need to consider the possibility of a non-real solution. In this case, you may need to use complex numbers to solve the equation.
Q: Can a radical equation have more than one solution?
A: Yes, a radical equation can have more than one solution. This can occur when the equation has multiple values that satisfy the equation.
Q: How do I determine if a radical equation has a solution?
A: To determine if a radical equation has a solution, you need to check if the expression under the radical sign is non-negative. If it is non-negative, then the equation has a solution.
Q: What if the expression under the radical sign is negative?
A: If the expression under the radical sign is negative, then the equation has no solution.
Q: Can a radical equation have a non-real solution?
A: Yes, a radical equation can have a non-real solution. This can occur when the equation has a negative number under the radical sign.
Q: How do I solve a radical equation with a non-real solution?
A: To solve a radical equation with a non-real solution, you need to use complex numbers. This involves using the imaginary unit, i, to represent the non-real solution.
Q: What is the imaginary unit, i?
A: The imaginary unit, i, is a complex number that is defined as the square root of -1. It is used to represent non-real solutions to equations.
Q: How do I use the imaginary unit, i, to solve a radical equation?
A: To use the imaginary unit, i, to solve a radical equation, you need to substitute i for the square root of -1 and then simplify the equation.
Q: Can a radical equation have a rational solution?
A: Yes, a radical equation can have a rational solution. This can occur when the expression under the radical sign is a perfect square.
Q: How do I determine if a radical equation has a rational solution?
A: To determine if a radical equation has a rational solution, you need to check if the expression under the radical sign is a perfect square.
Q: What is a perfect square?
A: A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be expressed as 2^2.
Q: How do I solve a radical equation with a rational solution?
A: To solve a radical equation with a rational solution, you need to take the square root of the expression under the radical sign and then simplify the equation.
Q: Can a radical equation have a mixed solution?
A: Yes, a radical equation can have a mixed solution. This can occur when the equation has both real and non-real solutions.
Q: How do I determine if a radical equation has a mixed solution?
A: To determine if a radical equation has a mixed solution, you need to check if the expression under the radical sign is both non-negative and non-real.
Q: What is the difference between a real solution and a non-real solution?
A: A real solution is a solution that can be expressed as a real number, while a non-real solution is a solution that cannot be expressed as a real number.
Q: Can a radical equation have a solution that is both real and non-real?
A: No, a radical equation cannot have a solution that is both real and non-real. A solution must be either real or non-real, but not both.
Q: How do I determine if a radical equation has a solution that is both real and non-real?
A: To determine if a radical equation has a solution that is both real and non-real, you need to check if the expression under the radical sign is both non-negative and non-real.
Q: What if the expression under the radical sign is neither non-negative nor non-real?
A: If the expression under the radical sign is neither non-negative nor non-real, then the equation has no solution.
Q: Can a radical equation have a solution that is a complex number?
A: Yes, a radical equation can have a solution that is a complex number. This can occur when the equation has a non-real solution.
Q: How do I determine if a radical equation has a solution that is a complex number?
A: To determine if a radical equation has a solution that is a complex number, you need to check if the expression under the radical sign is non-real.
Q: What is a complex number?
A: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
Q: How do I solve a radical equation with a complex solution?
A: To solve a radical equation with a complex solution, you need to use complex numbers. This involves using the imaginary unit, i, to represent the non-real solution.
Q: Can a radical equation have a solution that is a rational number?
A: Yes, a radical equation can have a solution that is a rational number. This can occur when the expression under the radical sign is a perfect square.
Q: How do I determine if a radical equation has a solution that is a rational number?
A: To determine if a radical equation has a solution that is a rational number, you need to check if the expression under the radical sign is a perfect square.
Q: What is a rational number?
A: A rational number is a number that can be expressed as the ratio of two integers. For example, 3/4 is a rational number.
Q: How do I solve a radical equation with a rational solution?
A: To solve a radical equation with a rational solution, you need to take the square root of the expression under the radical sign and then simplify the equation.
Q: Can a radical equation have a solution that is a mixed number?
A: Yes, a radical equation can have a solution that is a mixed number. This can occur when the equation has both real and non-real solutions.
Q: How do I determine if a radical equation has a solution that is a mixed number?
A: To determine if a radical equation has a solution that is a mixed number, you need to check if the expression under the radical sign is both non-negative and non-real.
Q: What is a mixed number?
A: A mixed number is a number that can be expressed as the sum of a whole number and a fraction. For example, 2 3/4 is a mixed number.
Q: How do I solve a radical equation with a mixed solution?
A: To solve a radical equation with a mixed solution, you need to use complex numbers. This involves using the imaginary unit, i, to represent the non-real solution.
Q: Can a radical equation have a solution that is a decimal number?
A: Yes, a radical equation can have a solution that is a decimal number. This can occur when the equation has a non-real solution.
Q: How do I determine if a radical equation has a solution that is a decimal number?
A: To determine if a radical equation has a solution that is a decimal number, you need to check if the expression under the radical sign is non-real.
Q: What is a decimal number?
A: A decimal number is a number that can be expressed as a decimal point followed by one or more digits. For example, 3.14 is a decimal number.
Q: How do I solve a radical equation with a decimal solution?
A: To solve a radical equation with a decimal solution, you need to use complex numbers. This involves using the imaginary unit, i, to represent the non-real solution.
Q: Can a radical equation have a solution that is an integer?
A: Yes, a radical equation can have a solution that is an integer. This can occur when the equation has a real solution.