Work Out The Value Of A A A Given The Equation:${ \sqrt{5^2 + 12^2} = \sqrt[3]{64 A^3} }$Your Final Line Should Say, A = … A = \ldots A = …

Introduction

In this article, we will delve into solving for the value of $a$ in the given equation $\sqrt{5^2 + 12^2} = \sqrt[3]{64 a^3}$. This equation involves both square and cube roots, making it a challenging problem to solve. We will break down the solution step by step, using algebraic manipulations and properties of radicals to isolate the variable $a$.

Step 1: Simplify the Left-Hand Side

The left-hand side of the equation involves the square root of the sum of two squares. We can simplify this expression by evaluating the sum inside the square root.

${ \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 }$

So, the left-hand side of the equation simplifies to $13$.

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation involves the cube root of $64a^3$. We can simplify this expression by evaluating the cube root.

${ \sqrt[3]{64 a^3} = \sqrt[3]{64} \cdot \sqrt[3]{a^3} = 4a }$

So, the right-hand side of the equation simplifies to $4a$.

Step 3: Equate the Simplified Expressions

Now that we have simplified both sides of the equation, we can equate them to each other.

${ 13 = 4a }$

Step 4: Solve for $a$

To solve for $a$, we can divide both sides of the equation by $4$.

${ a = \frac{13}{4} }$

Therefore, the value of $a$ is $\boxed{\frac{13}{4}}$.

Conclusion

In this article, we solved for the value of $a$ in the given equation $\sqrt{5^2 + 12^2} = \sqrt[3]{64 a^3}$. We simplified both sides of the equation using algebraic manipulations and properties of radicals, and then equated the simplified expressions to each other. Finally, we solved for $a$ by dividing both sides of the equation by $4$. The value of $a$ is $\frac{13}{4}$.

Additional Tips and Tricks

• When working with radicals, it's often helpful to simplify the expression inside the radical first.
• When simplifying expressions involving radicals, look for opportunities to use properties of radicals, such as the product rule and the quotient rule.
• When solving equations involving radicals, try to isolate the radical expression on one side of the equation, and then simplify the expression using algebraic manipulations.

Common Mistakes to Avoid

• When simplifying expressions involving radicals, be careful not to forget to simplify the expression inside the radical.
• When solving equations involving radicals, be careful not to forget to isolate the radical expression on one side of the equation.
• When working with radicals, be careful not to make mistakes when simplifying expressions or solving equations.
  Frequently Asked Questions (FAQs) about Solving for $a$ in the Given Equation ====================================================================================

Q: What is the given equation?

A: The given equation is $\sqrt{5^2 + 12^2} = \sqrt[3]{64 a^3}$.

Q: How do I simplify the left-hand side of the equation?

A: To simplify the left-hand side of the equation, you need to evaluate the sum inside the square root. In this case, the sum is $5^2 + 12^2$, which simplifies to $25 + 144 = 169$. Therefore, the left-hand side of the equation simplifies to $\sqrt{169} = 13$.

Q: How do I simplify the right-hand side of the equation?

A: To simplify the right-hand side of the equation, you need to evaluate the cube root of $64a^3$. In this case, the cube root of $64$ is $4$, and the cube root of $a^3$ is $a$. Therefore, the right-hand side of the equation simplifies to $4a$.

Q: How do I equate the simplified expressions?

A: To equate the simplified expressions, you need to set the left-hand side of the equation equal to the right-hand side of the equation. In this case, you have $13 = 4a$.

Q: How do I solve for $a$?

A: To solve for $a$, you need to isolate the variable $a$ on one side of the equation. In this case, you can divide both sides of the equation by $4$ to get $a = \frac{13}{4}$.

Q: What are some common mistakes to avoid when solving for $a$?

A: Some common mistakes to avoid when solving for $a$ include:

• Forgetting to simplify the expression inside the radical
• Forgetting to isolate the radical expression on one side of the equation
• Making mistakes when simplifying expressions or solving equations

Q: What are some additional tips and tricks for solving for $a$?

A: Some additional tips and tricks for solving for $a$ include:

• Simplifying the expression inside the radical first
• Using properties of radicals, such as the product rule and the quotient rule
• Isolating the radical expression on one side of the equation

Q: Can I use a calculator to solve for $a$?

A: Yes, you can use a calculator to solve for $a$. However, it's often helpful to understand the steps involved in solving the equation by hand, as this can help you to avoid mistakes and to understand the underlying math.

Q: How do I check my answer?

A: To check your answer, you can plug the value of $a$ back into the original equation and see if it's true. In this case, you can plug in $a = \frac{13}{4}$ and see if $\sqrt{5^2 + 12^2} = \sqrt[3]{64 a^3}$ is true.

Q: What if I get a different answer?

A: If you get a different answer, it's possible that you made a mistake somewhere in the process. Double-check your work and make sure that you followed the steps correctly. If you're still having trouble, you can try asking a teacher or tutor for help.