What Value Of { K $}$ Makes The Equation True?${ \left(5 A^2 B^3\right)\left(6 A^4 B\right)=30 A^6 B^4 }$A. 2 B. 3 C. 4 D. 8

Mar 8, 2025 by ADMIN 129 views

**Solving for the Value of k in the Given Equation**

Introduction

In this article, we will delve into the world of algebra and explore the concept of solving for the value of a variable in a given equation. Specifically, we will examine the equation ${\left(5 a^2 b^3\right)\left(6 a^4 b\right)=30 a^6 b^4}$ and determine the value of k that makes the equation true.

Understanding the Equation

The given equation is a product of two expressions, each containing variables a and b. To solve for the value of k, we need to simplify the equation and isolate the variable.

Simplifying the Equation

To simplify the equation, we can start by multiplying the two expressions together. Using the distributive property, we can rewrite the equation as:

${\left(5 a^2 b^3\right)\left(6 a^4 b\right) = 5 \cdot 6 \cdot a^2 \cdot a^4 \cdot b^3 \cdot b}$

Applying the Laws of Exponents

Next, we can apply the laws of exponents to simplify the equation further. Specifically, we can use the rule that states when multiplying two powers with the same base, we can add the exponents. In this case, we have:

${5 \cdot 6 \cdot a^2 \cdot a^4 \cdot b^3 \cdot b = 30 a^{2+4} b^{3+1}}$

Simplifying the Exponents

Using the rule mentioned above, we can simplify the exponents as follows:

${30 a^{2+4} b^{3+1} = 30 a^6 b^4}$

Comparing the Simplified Equation

Now that we have simplified the equation, we can compare it to the original equation to determine the value of k. Specifically, we can see that the simplified equation is equal to the original equation, which means that the value of k is equal to the exponent of a in the simplified equation.

Determining the Value of k

Based on the simplified equation, we can see that the exponent of a is 6. Therefore, the value of k is equal to 6.

Conclusion

In this article, we have solved for the value of k in the given equation. By simplifying the equation and applying the laws of exponents, we were able to determine that the value of k is equal to 6. This demonstrates the importance of simplifying equations and applying mathematical rules to solve for unknown values.

Final Answer

The final answer is: 6

Step-by-Step Solution

Multiply the two expressions together using the distributive property.
Apply the laws of exponents to simplify the equation.
Simplify the exponents by adding the exponents of the same base.
Compare the simplified equation to the original equation to determine the value of k.
The value of k is equal to the exponent of a in the simplified equation.

Key Concepts

Distributive property
Laws of exponents
Simplifying exponents
Comparing simplified equations to original equations

Mathematical Rules

When multiplying two powers with the same base, we can add the exponents.
When simplifying exponents, we can add the exponents of the same base.

Real-World Applications

Solving for the value of k in the given equation has real-world applications in fields such as physics, engineering, and computer science.
Understanding the laws of exponents and simplifying equations is crucial in solving complex mathematical problems.

Common Mistakes

Failing to apply the laws of exponents when simplifying equations.
Not comparing the simplified equation to the original equation to determine the value of k.

Tips and Tricks

When simplifying equations, always apply the laws of exponents.
When comparing simplified equations to original equations, make sure to check the exponents of the same base.
Q&A: Solving for the Value of k in the Given Equation =====================================================

Introduction

In our previous article, we solved for the value of k in the given equation ${\left(5 a^2 b^3\right)\left(6 a^4 b\right)=30 a^6 b^4}$ . In this article, we will answer some frequently asked questions related to the solution.

Q: What is the value of k in the given equation?

A: The value of k is equal to 6.

Q: How do I simplify the equation?

A: To simplify the equation, you can multiply the two expressions together using the distributive property, and then apply the laws of exponents to simplify the equation.

Q: What are the laws of exponents?

A: The laws of exponents are a set of rules that govern the behavior of exponents in mathematical operations. Specifically, when multiplying two powers with the same base, we can add the exponents.

Q: How do I apply the laws of exponents?

A: To apply the laws of exponents, you can simply add the exponents of the same base. For example, in the equation ${a^2 \cdot a^4 = a^{2+4} = a^6}$ , we added the exponents 2 and 4 to get the exponent 6.

Q: What is the importance of simplifying equations?

A: Simplifying equations is crucial in solving complex mathematical problems. By simplifying equations, we can make them easier to understand and solve, and we can also identify patterns and relationships that may not be apparent in the original equation.

Q: How do I compare the simplified equation to the original equation?

A: To compare the simplified equation to the original equation, you can simply check the exponents of the same base. If the exponents are the same, then the simplified equation is equal to the original equation.

Q: What are some common mistakes to avoid when solving for the value of k?

A: Some common mistakes to avoid when solving for the value of k include failing to apply the laws of exponents when simplifying equations, and not comparing the simplified equation to the original equation to determine the value of k.

Q: What are some real-world applications of solving for the value of k?

A: Solving for the value of k has real-world applications in fields such as physics, engineering, and computer science. For example, in physics, we may need to solve for the value of k in an equation that describes the motion of an object.

Q: How can I practice solving for the value of k?

A: You can practice solving for the value of k by working through examples and exercises that involve simplifying equations and applying the laws of exponents. You can also try solving for the value of k in different types of equations, such as linear and quadratic equations.

Additional Resources

For more information on the laws of exponents, see the article "Laws of Exponents: A Comprehensive Guide".
For more examples and exercises on solving for the value of k, see the article "Solving for the Value of k: Examples and Exercises".