Which Of The Following Satisfies The Equation Below?${ -5 \sqrt[3]{x+5} = -45 }$

Introduction

In this article, we will delve into solving the given equation, -5 ∛(x+5) = -45. This equation involves a cube root, which can be solved using algebraic manipulations. We will break down the solution step by step, making it easy to understand and follow.

Understanding the Equation

The given equation is -5 ∛(x+5) = -45. To solve for x, we need to isolate the variable x. The first step is to get rid of the cube root by cubing both sides of the equation.

Step 1: Cube Both Sides of the Equation

To eliminate the cube root, we will cube both sides of the equation. This will give us:

${ (-5 \sqrt[3]{x+5})^3 = (-45)^3 }$

Step 3: Simplify the Equation

Now, let's simplify the equation by expanding the cubes:

${ (-5)^3 (\sqrt[3]{x+5})^3 = (-45)^3 }$

Step 4: Simplify Further

Simplifying further, we get:

${ -125(x+5) = -91125 }$

Step 5: Distribute the Negative 125

Next, we will distribute the negative 125 to the terms inside the parentheses:

${ -125x - 625 = -91125 }$

Step 6: Add 625 to Both Sides

To isolate the term with the variable x, we will add 625 to both sides of the equation:

${ -125x = -91125 + 625 }$

Step 7: Simplify the Right Side

Simplifying the right side, we get:

${ -125x = -90500 }$

Step 8: Divide Both Sides by -125

Finally, we will divide both sides of the equation by -125 to solve for x:

${ x = \frac{-90500}{-125} }$

Step 9: Simplify the Fraction

Simplifying the fraction, we get:

${ x = 724 }$

Conclusion

In this article, we solved the equation -5 ∛(x+5) = -45 by cubing both sides of the equation and simplifying the resulting expression. We then isolated the variable x and solved for its value. The final solution is x = 724.

Frequently Asked Questions

• Q: What is the cube root of a number? A: The cube root of a number is a value that, when multiplied by itself twice, gives the original number.
• Q: How do I solve an equation with a cube root? A: To solve an equation with a cube root, you can cube both sides of the equation to eliminate the cube root.
• Q: What is the value of x in the equation -5 ∛(x+5) = -45? A: The value of x in the equation -5 ∛(x+5) = -45 is 724.

Additional Resources

• For more information on solving equations with cube roots, see the Khan Academy video on "Solving Equations with Cube Roots".
• For more practice problems on solving equations with cube roots, see the Mathway website.

Final Thoughts

Solving equations with cube roots can be challenging, but with practice and patience, you can master this skill. Remember to always cube both sides of the equation to eliminate the cube root, and then simplify the resulting expression to solve for the variable. With this knowledge, you will be able to tackle even the most complex equations with confidence.

Introduction

In our previous article, we solved the equation -5 ∛(x+5) = -45 by cubing both sides of the equation and simplifying the resulting expression. However, we know that there are many more questions and concerns that readers may have when it comes to solving equations with cube roots. In this article, we will address some of the most frequently asked questions and provide detailed answers to help you better understand this topic.

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27.

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

A: To solve an equation with a cube root, you can cube both sides of the equation to eliminate the cube root. This is because cubing both sides of the equation will cancel out the cube root, allowing you to solve for the variable.

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 twice, gives the original number, while a square root is a value that, when multiplied by itself, gives the original number. For example, the cube root of 27 is 3, while the square root of 16 is 4.

Q: Can I use a calculator to solve equations with cube roots?

A: Yes, you can use a calculator to solve equations with cube roots. However, it's always a good idea to understand the underlying math and to check your work to ensure that you have the correct solution.

Q: How do I simplify an expression with a cube root?

A: To simplify an expression with a cube root, you can use the following steps:

1. Factor out any perfect cubes from the expression.
2. Simplify the expression by canceling out any common factors.
3. Use the cube root property to rewrite the expression in a simpler form.

Q: Can I use the cube root property to solve equations with variables?

A: Yes, you can use the cube root property to solve equations with variables. The cube root property states that if a = b^3, then a^(1/3) = b. You can use this property to rewrite the equation in a simpler form and then solve for the variable.

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

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

• Not cubing both sides of the equation to eliminate the cube root.
• Not simplifying the expression by canceling out any common factors.
• Not using the cube root property to rewrite the equation in a simpler form.
• Not checking your work to ensure that you have the correct solution.

Q: How do I check my work when solving equations with cube roots?

A: To check your work when solving equations with cube roots, you can use the following steps:

1. Plug your solution back into the original equation to ensure that it is true.
2. Simplify the expression by canceling out any common factors.
3. Use the cube root property to rewrite the equation in a simpler form.
4. Check your work by plugging your solution back into the original equation.

Conclusion

Solving equations with cube roots can be challenging, but with practice and patience, you can master this skill. By understanding the cube root property and using it to simplify expressions, you can solve equations with cube roots with confidence. Remember to always check your work to ensure that you have the correct solution.

Additional Resources

• For more information on solving equations with cube roots, see the Khan Academy video on "Solving Equations with Cube Roots".
• For more practice problems on solving equations with cube roots, see the Mathway website.
• For a list of common mistakes to avoid when solving equations with cube roots, see the article on "Common Mistakes to Avoid When Solving Equations with Cube Roots".

Final Thoughts

Solving equations with cube roots is an important skill that can be used in a variety of mathematical contexts. By understanding the cube root property and using it to simplify expressions, you can solve equations with cube roots with confidence. Remember to always check your work to ensure that you have the correct solution. With practice and patience, you can master this skill and become proficient in solving equations with cube roots.