Solve For { X $}$ In The Equation: ${ X^2 \cdot X^7 = X }$
Introduction
When it comes to solving equations involving exponents, it's essential to understand the properties of exponents and how to manipulate them to isolate the variable. In this article, we will delve into the equation x^2 * x^7 = x and solve for x. We will use the properties of exponents, such as the product of powers property, to simplify the equation and isolate the variable.
Understanding the Equation
The given equation is x^2 * x^7 = x. To solve for x, we need to simplify the left-hand side of the equation using the properties of exponents. The product of powers property states that when we multiply two powers with the same base, we add the exponents. In this case, we have x^2 * x^7, which can be simplified to x^(2+7) = x^9.
Simplifying the Equation
Now that we have simplified the left-hand side of the equation to x^9, we can rewrite the original equation as x^9 = x. To solve for x, we need to isolate the variable. One way to do this is to use the property of equality, which states that if two expressions are equal, then we can perform the same operation on both sides of the equation.
Using the Property of Equality
Using the property of equality, we can subtract x from both sides of the equation x^9 = x to get x^9 - x = 0. This equation can be factored as x(x^8 - 1) = 0.
Factoring the Equation
The equation x(x^8 - 1) = 0 can be factored further using the difference of squares property. The difference of squares property states that a^2 - b^2 = (a - b)(a + b). In this case, we have x^8 - 1, which can be factored as (x^4 - 1)(x^4 + 1).
Factoring the Difference of Squares
The equation x(x^4 - 1)(x^4 + 1) = 0 can be factored further using the difference of squares property. The difference of squares property states that a^2 - b^2 = (a - b)(a + b). In this case, we have x^4 - 1, which can be factored as (x^2 - 1)(x^2 + 1).
Factoring the Difference of Squares Again
The equation x(x^2 - 1)(x^2 + 1)(x^4 + 1) = 0 can be factored further using the difference of squares property. The difference of squares property states that a^2 - b^2 = (a - b)(a + b). In this case, we have x^2 - 1, which can be factored as (x - 1)(x + 1).
Solving for x
Now that we have factored the equation x(x^2 - 1)(x^2 + 1)(x^4 + 1) = 0, we can solve for x. To do this, we need to set each factor equal to zero and solve for x.
Setting Each Factor Equal to Zero
Setting each factor equal to zero, we get:
- x = 0
- x^2 - 1 = 0
- x^2 + 1 = 0
- x^4 + 1 = 0
Solving for x in Each Factor
Solving for x in each factor, we get:
- x = 0
- x^2 - 1 = 0 --> x = ±1
- x^2 + 1 = 0 --> x = ±i
- x^4 + 1 = 0 --> x = ±i, ±(-i)
Conclusion
In conclusion, we have solved for x in the equation x^2 * x^7 = x. We used the properties of exponents, such as the product of powers property, to simplify the left-hand side of the equation. We then used the property of equality to isolate the variable and factored the equation using the difference of squares property. Finally, we solved for x in each factor and obtained the solutions x = 0, x = ±1, x = ±i, and x = ±(-i).
Final Answer
The final answer is:
Introduction
In our previous article, we solved for x in the equation x^2 * x^7 = x. We used the properties of exponents, such as the product of powers property, to simplify the left-hand side of the equation. We then used the property of equality to isolate the variable and factored the equation using the difference of squares property. Finally, we solved for x in each factor and obtained the solutions x = 0, x = ±1, x = ±i, and x = ±(-i). In this article, we will answer some frequently asked questions about solving for x in the equation x^2 * x^7 = x.
Q&A
Q: What is the product of powers property?
A: The product of powers property states that when we multiply two powers with the same base, we add the exponents. In this case, we have x^2 * x^7, which can be simplified to x^(2+7) = x^9.
Q: How do we simplify the left-hand side of the equation x^2 * x^7 = x?
A: We can simplify the left-hand side of the equation x^2 * x^7 = x by using the product of powers property. This property states that when we multiply two powers with the same base, we add the exponents. In this case, we have x^2 * x^7, which can be simplified to x^(2+7) = x^9.
Q: How do we use the property of equality to isolate the variable?
A: We can use the property of equality to isolate the variable by performing the same operation on both sides of the equation. In this case, we can subtract x from both sides of the equation x^9 = x to get x^9 - x = 0.
Q: How do we factor the equation x^9 - x = 0?
A: We can factor the equation x^9 - x = 0 by using the difference of squares property. The difference of squares property states that a^2 - b^2 = (a - b)(a + b). In this case, we have x^9 - x, which can be factored as x(x^8 - 1).
Q: How do we factor the equation x(x^8 - 1) = 0?
A: We can factor the equation x(x^8 - 1) = 0 by using the difference of squares property. The difference of squares property states that a^2 - b^2 = (a - b)(a + b). In this case, we have x^8 - 1, which can be factored as (x^4 - 1)(x^4 + 1).
Q: How do we solve for x in each factor?
A: We can solve for x in each factor by setting each factor equal to zero and solving for x. In this case, we have x = 0, x^2 - 1 = 0, x^2 + 1 = 0, and x^4 + 1 = 0.
Q: What are the solutions to the equation x^2 * x^7 = x?
A: The solutions to the equation x^2 * x^7 = x are x = 0, x = ±1, x = ±i, and x = ±(-i).
Conclusion
In conclusion, we have answered some frequently asked questions about solving for x in the equation x^2 * x^7 = x. We used the properties of exponents, such as the product of powers property, to simplify the left-hand side of the equation. We then used the property of equality to isolate the variable and factored the equation using the difference of squares property. Finally, we solved for x in each factor and obtained the solutions x = 0, x = ±1, x = ±i, and x = ±(-i).
Final Answer
The final answer is: