Simplifying (2m³n²)⁵ A Step-by-Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill, especially when dealing with exponents. Exponents, or powers, represent repeated multiplication, and mastering their manipulation is crucial for success in algebra and beyond. This article will delve into the process of simplifying the expression ${2 m^3 n^2\}^5$, providing a step-by-step guide and highlighting the underlying principles.

Understanding the Basics of Exponents

Before we dive into the specifics of simplifying ${2 m^3 n^2\}^5$, let's recap the basic rules of exponents. An exponent indicates how many times a base is multiplied by itself. For instance, $x^3$ means x multiplied by itself three times (x * x * x). When dealing with expressions involving exponents, several key rules come into play:

  • Product of Powers: When multiplying exponents with the same base, you add the powers. For example, $x^m * x^n = x^{m+n}$.
  • Quotient of Powers: When dividing exponents with the same base, you subtract the powers. For example, $x^m / x^n = x^{m-n}$.
  • Power of a Power: When raising a power to another power, you multiply the exponents. For example, $(xm)n = x^{m*n}$.
  • Power of a Product: When raising a product to a power, you apply the power to each factor in the product. For example, $(xy)^n = x^n * y^n$.
  • Power of a Quotient: When raising a quotient to a power, you apply the power to both the numerator and the denominator. For example, $(x/y)^n = x^n / y^n$.
  • Zero Exponent: Any non-zero number raised to the power of zero equals 1. For example, $x^0 = 1$ (where x ≠ 0).
  • Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, $x^{-n} = 1/x^n$.

These rules form the foundation for simplifying exponential expressions. By understanding and applying these rules, we can efficiently manipulate complex expressions into their simplest forms.

Step-by-Step Simplification of (2m³n²)⁵

Now, let's tackle the expression ${2 m^3 n^2\}^5$ step by step. Our goal is to simplify this expression by applying the rules of exponents. Here's how we'll proceed:

Step 1: Applying the Power of a Product Rule

The first step is to recognize that we have a product inside the parentheses being raised to a power. The power of a product rule states that $(xy)^n = x^n * y^n$. Applying this rule to our expression, we get:

(2m3n2)5=25∗(m3)5∗(n2)5(2 m^3 n^2)^5 = 2^5 * (m^3)^5 * (n^2)^5

This step distributes the exponent of 5 to each factor within the parentheses: the constant 2, the variable $m^3$, and the variable $n^2$. We've effectively broken down the original expression into smaller, more manageable parts. This is a crucial step in simplifying complex expressions, as it allows us to apply other exponent rules more easily.

Step 2: Applying the Power of a Power Rule

Next, we encounter terms where a power is raised to another power. This is where the power of a power rule comes into play, which states that $(xm)n = x^{m*n}$. We have two such terms in our expression: $(m3)5$ and $(n2)5$. Applying the power of a power rule to these terms, we multiply the exponents:

  • (m3)5=m3∗5=m15(m^3)^5 = m^{3*5} = m^{15}

  • (n2)5=n2∗5=n10(n^2)^5 = n^{2*5} = n^{10}

This step further simplifies the expression by reducing the nested exponents to single exponents. We've now eliminated the parentheses around the variable terms, making the expression even more streamlined. Understanding and applying the power of a power rule is essential for simplifying expressions with multiple exponents.

Step 3: Evaluating the Constant Term

We also have the term $2^5$, which represents 2 raised to the power of 5. This is a straightforward calculation:

25=2∗2∗2∗2∗2=322^5 = 2 * 2 * 2 * 2 * 2 = 32

Evaluating the constant term is a crucial step in fully simplifying the expression. This step converts the exponential form of the constant into its numerical value, providing a more concise representation.

Step 4: Combining the Simplified Terms

Now that we've simplified each part of the expression, we can combine the results:

25∗(m3)5∗(n2)5=32∗m15∗n102^5 * (m^3)^5 * (n^2)^5 = 32 * m^{15} * n^{10}

This step brings together all the simplified components into a single expression. We've replaced the exponential terms with their simplified forms, resulting in a clear and concise representation of the original expression. Combining the simplified terms is the final step in the simplification process.

Final Simplified Expression

Therefore, the simplified form of ${2 m^3 n^2\}^5$ is:

32m15n1032m^{15}n^{10}

This is the final simplified form of the given expression. We've successfully applied the rules of exponents to transform the original expression into a more compact and understandable form. This simplified expression is easier to work with in further mathematical operations.

Common Mistakes to Avoid

When simplifying exponential expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Forgetting to Apply the Power to All Factors: A common mistake is to only apply the outer exponent to some factors inside the parentheses, but not all. Remember, the power of a product rule requires you to apply the exponent to every factor. For example, in ${2 m^3 n^2\}^5$, the exponent 5 must be applied to 2, $m^3$, and $n^2$. Failing to do so will lead to an incorrect simplification.
  • Incorrectly Adding Exponents When Multiplying Powers: When multiplying powers with the same base, you add the exponents. However, it's a mistake to add exponents when the bases are different. For instance, $x^2 * y^3$ cannot be simplified further by adding the exponents because the bases (x and y) are different. Only exponents with the same base can be added during multiplication.
  • Incorrectly Multiplying Exponents When Raising a Power to a Power: The power of a power rule states that you multiply exponents when raising a power to another power. A common mistake is to add the exponents instead. For example, $(x3)2$ should be simplified as $x^{3*2} = x^6$, not $x^{3+2} = x^5$. Always remember to multiply the exponents in this scenario.
  • Ignoring the Order of Operations: The order of operations (PEMDAS/BODMAS) is crucial in simplifying any mathematical expression, including those with exponents. Make sure to address parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right) in the correct order. Ignoring this order can lead to incorrect results.
  • Misunderstanding Negative Exponents: Negative exponents indicate reciprocals. For example, $x^{-2} = 1/x^2$. A common mistake is to treat a negative exponent as a negative number. Remember that a negative exponent means the base is in the denominator with a positive exponent.
  • Forgetting the Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1. This rule is often overlooked, leading to errors in simplification. For example, $5^0 = 1$, not 0.

By being aware of these common mistakes, you can significantly improve your accuracy in simplifying exponential expressions. Double-checking your work and paying close attention to the rules of exponents are essential for success.

Practice Problems

To solidify your understanding of simplifying exponential expressions, here are a few practice problems:

  1. Simplify ${3x^2y^4\}^3$
  2. Simplify $\frac{10a5b3}{2a^2b}$
  3. Simplify ${4p^{-2}q^5\}^{-2}$
  4. Simplify $(x2y3)(x4y{-1})$
  5. Simplify $\left(\frac{2m4n}{m2n3}\right)2$

Working through these problems will help you apply the rules of exponents in different contexts and build your confidence in simplifying expressions. Remember to break down each problem into smaller steps, apply the appropriate rules, and double-check your work.

Conclusion

Simplifying exponential expressions is a crucial skill in mathematics. By understanding and applying the rules of exponents, you can efficiently manipulate complex expressions into their simplest forms. In this article, we've walked through the step-by-step simplification of ${2 m^3 n^2\}^5$, highlighted common mistakes to avoid, and provided practice problems to further your understanding. With consistent practice and a solid grasp of the fundamental principles, you can master the art of simplifying exponential expressions.

Remember, the key to success in mathematics is practice. The more you work with exponential expressions, the more comfortable and confident you'll become in simplifying them. So, keep practicing, keep learning, and you'll excel in your mathematical journey.