Solve For X X X : X 1 3 = 2 − 3 X^{\frac{1}{3}} = 2^{-3} X 3 1 ​ = 2 − 3

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Introduction

In mathematics, solving cubic equations is a crucial aspect of algebraic manipulation. These equations involve a variable raised to the power of 3, and solving them requires a deep understanding of exponent rules and algebraic techniques. In this article, we will focus on solving the cubic equation $x^{\frac{1}{3}} = 2^{-3}$, which involves a fractional exponent. We will break down the solution into manageable steps, using a combination of algebraic manipulation and exponent rules to find the value of $x$.

Understanding the Equation

The given equation is $x^{\frac{1}{3}} = 2^{-3}$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. The first step is to understand the properties of fractional exponents. A fractional exponent $\frac{1}{n}$ represents the $n$th root of a number. In this case, $x^{\frac{1}{3}}$ represents the cube root of $x$.

Step 1: Simplify the Right-Hand Side

The right-hand side of the equation is $2^{-3}$. To simplify this expression, we can use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Applying this property, we get:

$$2^{-3} = \frac{1}{2^3}$$

Step 2: Evaluate the Expression

Now that we have simplified the right-hand side, we can evaluate the expression. The cube of 2 is 8, so we can write:

$$\frac{1}{2^3} = \frac{1}{8}$$

Step 3: Rewrite the Equation

Now that we have simplified the right-hand side, we can rewrite the equation as:

$$x^{\frac{1}{3}} = \frac{1}{8}$$

Step 4: Eliminate the Fractional Exponent

To eliminate the fractional exponent, we can raise both sides of the equation to the power of 3. This will cancel out the fractional exponent on the left-hand side:

$$(x^{\frac{1}{3}})^3 = \left(\frac{1}{8}\right)^3$$

Step 5: Simplify the Left-Hand Side

When we raise a power to another power, we multiply the exponents. In this case, we have:

$$(x^{\frac{1}{3}})^3 = x^{\frac{1}{3} \cdot 3} = x^1$$

Step 6: Simplify the Right-Hand Side

Now that we have simplified the left-hand side, we can simplify the right-hand side. When we raise a fraction to a power, we raise the numerator and denominator to that power. In this case, we have:

$$\left(\frac{1}{8}\right)^3 = \frac{1^3}{8^3} = \frac{1}{512}$$

Step 7: Solve for $x$

Now that we have simplified both sides of the equation, we can solve for $x$. We have:

$$x^1 = \frac{1}{512}$$

To solve for $x$, we can take the reciprocal of both sides of the equation:

$$x = \frac{1}{\frac{1}{512}} = 512$$

Conclusion

In this article, we solved the cubic equation $x^{\frac{1}{3}} = 2^{-3}$ using a combination of algebraic manipulation and exponent rules. We broke down the solution into manageable steps, simplifying the right-hand side and eliminating the fractional exponent on the left-hand side. Finally, we solved for $x$ by taking the reciprocal of both sides of the equation. The value of $x$ is 512.

Additional Tips and Tricks

• When solving cubic equations, it's essential to understand the properties of fractional exponents and how to manipulate them.
• Raising both sides of the equation to the power of 3 can help eliminate the fractional exponent on the left-hand side.
• Taking the reciprocal of both sides of the equation can help solve for $x$.

Common Mistakes to Avoid

• Failing to simplify the right-hand side of the equation can lead to incorrect solutions.
• Not eliminating the fractional exponent on the left-hand side can make it difficult to solve for $x$.
• Not taking the reciprocal of both sides of the equation can lead to incorrect solutions.

Real-World Applications

Solving cubic equations has numerous real-world applications, including:

• Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Cubic equations are used in algorithms and data structures, such as binary search trees.

Conclusion

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Introduction

In our previous article, we solved the cubic equation $x^{\frac{1}{3}} = 2^{-3}$ using a combination of algebraic manipulation and exponent rules. In this article, we will answer some of the most frequently asked questions about solving cubic equations.

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree 3, which means that the highest power of the variable is 3. Cubic equations can be written in the form $ax^3 + bx^2 + cx + d = 0$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you can use a combination of algebraic manipulation and exponent rules. Here are the steps:

1. Simplify the right-hand side of the equation.
2. Eliminate the fractional exponent on the left-hand side.
3. Raise both sides of the equation to the power of 3.
4. Simplify the left-hand side.
5. Solve for $x$.

Q: What is a fractional exponent?

A: A fractional exponent is a power that is not a whole number. For example, $\frac{1}{3}$ is a fractional exponent. When you raise a number to a fractional exponent, you are taking the root of the number.

Q: How do I eliminate a fractional exponent?

A: To eliminate a fractional exponent, you can raise both sides of the equation to the power of the denominator of the fractional exponent. For example, if you have $x^{\frac{1}{3}} = 2$, you can raise both sides to the power of 3 to eliminate the fractional exponent.

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

A: A quadratic equation is a polynomial equation of degree 2, which means that the highest power of the variable is 2. A cubic equation, on the other hand, is a polynomial equation of degree 3, which means that the highest power of the variable is 3.

Q: Can I use a calculator to solve a cubic equation?

A: Yes, you can use a calculator to solve a cubic equation. However, it's always a good idea to check your work by hand to make sure that the solution is correct.

Q: What are some real-world applications of cubic equations?

A: Cubic equations have numerous real-world applications, including:

• Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Cubic equations are used in algorithms and data structures, such as binary search trees.

Q: Can I use a cubic equation to model a real-world problem?

A: Yes, you can use a cubic equation to model a real-world problem. For example, you can use a cubic equation to model the motion of a projectile under the influence of gravity.

Q: How do I know if a cubic equation has a real solution?

A: To determine if a cubic equation has a real solution, you can use the discriminant. The discriminant is a value that is calculated from the coefficients of the equation. If the discriminant is positive, the equation has a real solution. If the discriminant is negative, the equation has no real solution.

Conclusion

In conclusion, solving cubic equations is a crucial aspect of algebraic manipulation. By understanding the properties of fractional exponents and using a combination of algebraic manipulation and exponent rules, we can solve complex equations like $x^{\frac{1}{3}} = 2^{-3}$. We hope that this Q&A guide has been helpful in answering some of the most frequently asked questions about solving cubic equations.