Use The Laws Of Exponents To Simplify. Do Not Use Negative Exponents.${\left(m^{\frac{3}{2}} N {-\frac{4}{3}}\right) {\frac{6}{7}}}$

Understanding the Laws of Exponents

The laws of exponents are a set of rules that help us simplify expressions involving exponents. These laws are essential in mathematics, particularly in algebra and calculus. In this article, we will focus on simplifying the given expression using the laws of exponents.

The Given Expression

The given expression is $\left(m^{\frac{3}{2}} n^{-\frac{4}{3}}\right)^{\frac{6}{7}}$. Our goal is to simplify this expression without using negative exponents.

Applying the Laws of Exponents

To simplify the given expression, we will apply the laws of exponents. The first law we will use is the power of a power law, which states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{mn}$.

Step 1: Apply the Power of a Power Law

Using the power of a power law, we can rewrite the given expression as:

$$\left(m^{\frac{3}{2}} n^{-\frac{4}{3}}\right)^{\frac{6}{7}} = m^{\frac{3}{2} \cdot \frac{6}{7}} n^{-\frac{4}{3} \cdot \frac{6}{7}}$$

Step 2: Simplify the Exponents

Now, we can simplify the exponents by multiplying the fractions:

$$m^{\frac{3}{2} \cdot \frac{6}{7}} n^{-\frac{4}{3} \cdot \frac{6}{7}} = m^{\frac{18}{14}} n^{-\frac{24}{21}}$$

Step 3: Simplify the Fractions

We can simplify the fractions by dividing both the numerator and the denominator by their greatest common divisor:

$$m^{\frac{18}{14}} n^{-\frac{24}{21}} = m^{\frac{9}{7}} n^{-\frac{8}{7}}$$

Step 4: Rewrite the Expression

Now, we can rewrite the expression using the simplified exponents:

$$m^{\frac{9}{7}} n^{-\frac{8}{7}} = \frac{m^{\frac{9}{7}}}{n^{\frac{8}{7}}}$$

Conclusion

In conclusion, we have successfully simplified the given expression using the laws of exponents. The final expression is $\frac{m^{\frac{9}{7}}}{n^{\frac{8}{7}}}$.

Importance of the Laws of Exponents

The laws of exponents are essential in mathematics, particularly in algebra and calculus. They help us simplify expressions involving exponents, which is crucial in solving mathematical problems. By applying the laws of exponents, we can rewrite expressions in a simpler form, making it easier to solve problems.

Real-World Applications

The laws of exponents have numerous real-world applications. For example, in physics, the laws of exponents are used to describe the behavior of physical systems, such as the motion of objects and the behavior of electrical circuits. In engineering, the laws of exponents are used to design and optimize systems, such as bridges and buildings.

Final Thoughts

In conclusion, the laws of exponents are a powerful tool in mathematics, particularly in algebra and calculus. By applying these laws, we can simplify expressions involving exponents, which is crucial in solving mathematical problems. The laws of exponents have numerous real-world applications, making them an essential part of mathematics.

Frequently Asked Questions

Q: What are the laws of exponents?

A: The laws of exponents are a set of rules that help us simplify expressions involving exponents.

Q: How do I apply the laws of exponents?

A: To apply the laws of exponents, you need to follow the rules of exponentiation, such as the power of a power law and the product of powers law.

Q: What are the real-world applications of the laws of exponents?

A: The laws of exponents have numerous real-world applications, including physics, engineering, and computer science.

Q: Why are the laws of exponents important?

A: The laws of exponents are essential in mathematics, particularly in algebra and calculus. They help us simplify expressions involving exponents, which is crucial in solving mathematical problems.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.