What Could Be The Value Of $x$ In The Following Equation? Select All That Apply.$x^3 = \frac{1}{64}$A. \$-\frac{1}{4}$[/tex\]B. $-\sqrt[3]{\frac{1}{64}}$C. $\frac{1}{4}$D.

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Introduction

In mathematics, solving equations is a fundamental concept that helps us understand the relationships between variables. One of the most common types of equations is the cubic equation, which involves a variable raised to the power of 3. In this article, we will explore the cubic equation $x^3 = \frac{1}{64}$ and determine the possible values of x.

Understanding the Equation

The given equation is $x^3 = \frac{1}{64}$. To solve for x, we need to isolate the variable x. Since the equation involves a cube root, we can take the cube root of both sides to get rid of the exponent.

Taking the Cube Root

Taking the cube root of both sides of the equation gives us:

x33=1643\sqrt[3]{x^3} = \sqrt[3]{\frac{1}{64}}

Using the property of cube roots, we can simplify the left-hand side of the equation to get:

x=1643x = \sqrt[3]{\frac{1}{64}}

Evaluating the Cube Root

Now, let's evaluate the cube root of $\frac{1}{64}$. We can rewrite $\frac{1}{64}$ as $\frac{1}{4^3}$, which is equivalent to $\left(\frac{1}{4}\right)^3$.

Using the property of exponents, we can rewrite the cube root as:

x=(14)13x = \left(\frac{1}{4}\right)^{\frac{1}{3}}

Simplifying the Expression

To simplify the expression, we can rewrite the cube root as:

x=143x = \frac{1}{\sqrt[3]{4}}

Evaluating the Cube Root of 4

The cube root of 4 is equal to $\sqrt[3]{4} = 1.5874$ (rounded to four decimal places).

Substituting the Value

Substituting the value of the cube root of 4 into the expression, we get:

x=11.5874x = \frac{1}{1.5874}

Simplifying the Fraction

To simplify the fraction, we can divide the numerator and denominator by their greatest common divisor, which is 1.

The Final Answer

The final answer is:

x=11.5874≈0.62996x = \frac{1}{1.5874} \approx 0.62996

Conclusion

In conclusion, the possible values of x in the equation $x^3 = \frac{1}{64}$ are:

  • x=−14x = -\frac{1}{4}

  • x=−1643x = -\sqrt[3]{\frac{1}{64}}

  • x=14x = \frac{1}{4}

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means that the highest power of the variable is three. It is typically 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 various methods such as factoring, the rational root theorem, or the cubic formula. However, for equations of the form $x^3 = k$, where k is a constant, you can simply take the cube root of both sides to get the solution.

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is denoted by the symbol $\sqrt[3]{x}$, where x is the number.

Q: How do I evaluate the cube root of a fraction?

A: To evaluate the cube root of a fraction, you can rewrite the fraction as a product of two numbers, one of which is a perfect cube. Then, you can take the cube root of each number separately and multiply the results.

Q: What is the difference between a cube root and a square root?

A: A cube root is a value that, when multiplied by itself three times, gives the original number, while a square root is a value that, when multiplied by itself twice, gives the original number. In other words, the cube root is the inverse operation of cubing, while the square root is the inverse operation of squaring.

Q: Can I simplify a cube root expression?

A: Yes, you can simplify a cube root expression by rewriting it as a product of two numbers, one of which is a perfect cube. Then, you can take the cube root of each number separately and multiply the results.

Q: How do I use the cube root to solve equations?

A: To use the cube root to solve equations, you can take the cube root of both sides of the equation to get rid of the exponent. This will give you a new equation that is easier to solve.

Q: What are some common mistakes to avoid when working with cube roots?

A: Some common mistakes to avoid when working with cube roots include:

  • Not simplifying the expression before taking the cube root
  • Not rewriting the fraction as a product of two numbers
  • Not taking the cube root of both sides of the equation
  • Not checking for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can plug the solution back into the original equation and see if it is true. If it is not true, then the solution is extraneous and should be discarded.

Q: What are some real-world applications of cube roots?

A: Cube roots have many real-world applications, including:

  • Physics: Cube roots are used to calculate the volume of a cube or the length of a side of a cube.
  • Engineering: Cube roots are used to calculate the stress on a material or the strain on a structure.
  • Computer Science: Cube roots are used in algorithms for solving equations and optimizing functions.

Conclusion

In conclusion, cube roots are an important concept in mathematics that have many real-world applications. By understanding how to evaluate and simplify cube root expressions, you can solve equations and optimize functions with ease.