Simplify The Expression: $\frac{4 M^3 N^2}{3 M^{-3}}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve problems more efficiently. It involves reducing complex expressions to their simplest form, making it easier to understand and work with. In this article, we will simplify the expression $\frac{4 m^3 n^2}{3 m^{-3}}$ using the rules of exponents and algebra.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $a^3$ means $a \times a \times a$. When we have a negative exponent, it means we are taking the reciprocal of the expression. For instance, $a^{-3}$ means $\frac{1}{a^3}$. In the given expression, we have a negative exponent in the denominator, which we will simplify using the rules of exponents.

Simplifying the Expression

To simplify the expression $\frac{4 m^3 n^2}{3 m^{-3}}$, we will use the rule of exponents that states when we divide two powers with the same base, we subtract the exponents. In this case, the base is $m$, and the exponents are $3$ and $-3$. So, we can simplify the expression as follows:

$$\frac{4 m^3 n^2}{3 m^{-3}} = \frac{4 m^{3-(-3)} n^2}{3}$$

Using the rule of exponents, we can simplify the exponent $3-(-3)$ as follows:

$$3-(-3) = 3 + 3 = 6$$

So, the expression becomes:

$$\frac{4 m^6 n^2}{3}$$

Simplifying the Numerator

The numerator of the expression is $4 m^6 n^2$. We can simplify this expression by combining the constants and the variables. The constant $4$ can be written as $4 \times 1$, and the variable $m^6$ can be written as $m^6 \times 1$. So, the numerator becomes:

$$4 \times 1 \times m^6 \times n^2$$

Using the rule of multiplication, we can combine the constants and the variables as follows:

$$4 \times 1 \times m^6 \times n^2 = 4 m^6 n^2$$

Simplifying the Expression Further

Now that we have simplified the numerator, we can simplify the expression further by combining the constants and the variables. The constant $3$ in the denominator can be written as $3 \times 1$, and the variable $m^6$ in the numerator can be written as $m^6 \times 1$. So, the expression becomes:

$$\frac{4 m^6 n^2}{3 \times 1}$$

Using the rule of division, we can simplify the expression as follows:

$$\frac{4 m^6 n^2}{3 \times 1} = \frac{4 m^6 n^2}{3}$$

Conclusion

In this article, we simplified the expression $\frac{4 m^3 n^2}{3 m^{-3}}$ using the rules of exponents and algebra. We used the rule of exponents to simplify the negative exponent in the denominator, and then we simplified the numerator by combining the constants and the variables. Finally, we simplified the expression further by combining the constants and the variables. The simplified expression is $\frac{4 m^6 n^2}{3}$.

Final Answer

The final answer is $\boxed{\frac{4 m^6 n^2}{3}}$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

• Exponent: A shorthand way of writing repeated multiplication.
• Negative Exponent: A way of writing the reciprocal of an expression.
• Rule of Exponents: A set of rules that govern the behavior of exponents.
• Simplifying an Expression: Reducing a complex expression to its simplest form.
  Simplify the Expression: $\frac{4 m^3 n^2}{3 m^{-3}}$ - Q&A ===========================================================

Introduction

In our previous article, we simplified the expression $\frac{4 m^3 n^2}{3 m^{-3}}$ using the rules of exponents and algebra. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q: What is the rule of exponents?

A: The rule of exponents states that when we divide two powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do we simplify a negative exponent?

A: To simplify a negative exponent, we take the reciprocal of the expression. For example, $a^{-n} = \frac{1}{a^n}$.

Q: Can we simplify the expression $\frac{4 m^3 n^2}{3 m^{-3}}$ further?

A: Yes, we can simplify the expression further by combining the constants and the variables. The constant $4$ in the numerator can be written as $4 \times 1$, and the variable $m^6$ in the numerator can be written as $m^6 \times 1$. So, the expression becomes:

$$\frac{4 \times 1 \times m^6 \times n^2}{3 \times 1}$$

Using the rule of division, we can simplify the expression as follows:

$$\frac{4 \times 1 \times m^6 \times n^2}{3 \times 1} = \frac{4 m^6 n^2}{3}$$

Q: What is the final answer to the expression $\frac{4 m^3 n^2}{3 m^{-3}}$?

A: The final answer to the expression $\frac{4 m^3 n^2}{3 m^{-3}}$ is $\boxed{\frac{4 m^6 n^2}{3}}$.

Q: Can we use the rule of exponents to simplify other expressions?

A: Yes, we can use the rule of exponents to simplify other expressions. For example, if we have the expression $\frac{a^m}{a^n}$, we can simplify it using the rule of exponents as follows:

$$\frac{a^m}{a^n} = a^{m-n}$$

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not using the rule of exponents correctly
• Not simplifying the numerator and denominator separately
• Not combining the constants and variables correctly
• Not checking the final answer for errors

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\frac{4 m^3 n^2}{3 m^{-3}}$. We discussed the rule of exponents, how to simplify negative exponents, and how to simplify the expression further by combining the constants and variables. We also provided some common mistakes to avoid when simplifying expressions.

Final Answer

The final answer to the expression $\frac{4 m^3 n^2}{3 m^{-3}}$ is $\boxed{\frac{4 m^6 n^2}{3}}$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

• Exponent: A shorthand way of writing repeated multiplication.
• Negative Exponent: A way of writing the reciprocal of an expression.
• Rule of Exponents: A set of rules that govern the behavior of exponents.
• Simplifying an Expression: Reducing a complex expression to its simplest form.