Rewriting Expressions With Exponent Properties A Step By Step Guide

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In mathematics, particularly in algebra, understanding and applying the properties of exponents is fundamental. Exponents, which denote the power to which a number is raised, follow specific rules that allow us to simplify complex expressions. This article will delve into the application of these properties, focusing on rewriting the expression $ rac{3 imes 3{\frac{2}{5}}}{9{\frac{7}{10}}}$ step-by-step. We will explore the laws of exponents, such as the product of powers, the power of a power, and the quotient of powers, to simplify this expression effectively. This exploration is not just an academic exercise; mastering these concepts is crucial for tackling more advanced mathematical problems in calculus, physics, and engineering. So, let's embark on this mathematical journey, breaking down the intricacies of exponents and their properties, and learn how to rewrite expressions with confidence and precision.

Understanding the Basics of Exponents

At its core, an exponent indicates how many times a base number is multiplied by itself. For instance, in the expression ana^n, ‘a’ is the base, and ‘n’ is the exponent. This means ‘a’ is multiplied by itself ‘n’ times. Understanding this fundamental concept is the bedrock for grasping more complex exponent properties. Now, let’s delve into the specific properties that govern how exponents behave under different operations.

Key Properties of Exponents

Several key properties dictate how exponents work. These properties are the tools we use to simplify and manipulate expressions involving exponents. Some of the most crucial ones include:

  1. Product of Powers: This property states that when multiplying two powers with the same base, you add the exponents. Mathematically, it's represented as amimesan=am+na^m imes a^n = a^{m+n}.
  2. Quotient of Powers: Conversely, when dividing two powers with the same base, you subtract the exponents. The formula is am/an=am−na^m / a^n = a^{m-n}.
  3. Power of a Power: This property deals with raising a power to another power. In this case, you multiply the exponents: (am)n=amimesn(a^m)^n = a^{m imes n}.
  4. Power of a Product: When a product is raised to a power, each factor in the product is raised to that power: (ab)n=anbn(ab)^n = a^n b^n.
  5. Power of a Quotient: Similar to the power of a product, when a quotient is raised to a power, both the numerator and the denominator are raised to that power: (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.
  6. Zero Exponent: Any non-zero number raised to the power of zero is one: a0=1a^0 = 1 (where a≠0a \neq 0).
  7. Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive of the exponent: a−n=1ana^{-n} = \frac{1}{a^n}.
  8. Fractional Exponents: Fractional exponents connect exponents with radicals. For example, a1na^{\frac{1}{n}} is the nth root of a, and amna^{\frac{m}{n}} is the nth root of ama^m. Understanding fractional exponents is crucial for simplifying expressions involving roots and exponents.

These properties are not just abstract rules; they are the keys to simplifying complex mathematical expressions. To truly master them, it's essential to see them in action. Let’s move on to the step-by-step application of these properties to rewrite the expression in question.

Step-by-Step Rewriting of the Expression

Now, let's apply these exponent properties to rewrite the expression: $ rac{3 imes 3{\frac{2}{5}}}{9{\frac{7}{10}}}$. We'll break this down step by step to make the process clear and understandable.

Step 1: Expressing the Base in Prime Factors

The first step in simplifying this expression is to express all bases in their prime factors. This is a crucial step because it allows us to apply the properties of exponents more easily. In our expression, we have two bases: 3 and 9. The number 3 is already a prime number, but 9 can be expressed as 323^2. Rewriting 9 as 323^2 allows us to have a common base throughout the expression, which is essential for applying exponent rules effectively. So, let's rewrite the expression, replacing 9 with 323^2. This gives us: $ rac{3 imes 3{\frac{2}{5}}}{(32)^{\frac{7}{10}}}$. Now that we have a common base, we can proceed to the next step, which involves applying the power of a power property.

Step 2: Applying the Power of a Power Property

The next step involves simplifying the denominator of the expression. We have (32)710(3^2)^{\frac{7}{10}} in the denominator, which is a power raised to another power. According to the power of a power property, (am)n=amimesn(a^m)^n = a^{m imes n}, we need to multiply the exponents. In this case, we multiply 2 by 710\frac{7}{10}. This multiplication simplifies as follows: $2 imes \frac7}{10} = \frac{14}{10} = \frac{7}{5}$. So, (32)710(3^2)^{\frac{7}{10}} simplifies to 3753^{\frac{7}{5}}. Our expression now looks like this $ rac{3 imes 3^{\frac{2{5}}}{3^{\frac{7}{5}}}$. With the denominator simplified, we can now focus on the numerator and apply the product of powers property.

Step 3: Applying the Product of Powers Property in the Numerator

Now, let's simplify the numerator. We have 3imes3253 imes 3^{\frac{2}{5}}. Remember that any number without an explicitly written exponent is understood to have an exponent of 1. So, we can rewrite 3 as 313^1. According to the product of powers property, amimesan=am+na^m imes a^n = a^{m+n}, we need to add the exponents when multiplying powers with the same base. In this case, we add the exponents 1 and 25\frac{2}{5}. To add these, we need a common denominator, so we rewrite 1 as 55\frac{5}{5}. The addition then becomes: $\frac5}{5} + \frac{2}{5} = \frac{7}{5}$. Therefore, 31imes3253^1 imes 3^{\frac{2}{5}} simplifies to 3753^{\frac{7}{5}}. Our expression now looks significantly simpler $\frac{3^{\frac{7{5}}}{3^{\frac{7}{5}}}$. The next step is to apply the quotient of powers property to fully simplify the expression.

Step 4: Applying the Quotient of Powers Property

We've reached a crucial point in simplifying the expression where we have a quotient of powers with the same base. Our expression is now: $\frac3{\frac{7}{5}}}{3{\frac{7}{5}}}$. This is where the quotient of powers property comes into play. This property states that when dividing two powers with the same base, we subtract the exponents $\frac{a^ma^n} = a^{m-n}$. In our case, the base is 3, and the exponents are both 75\frac{7}{5}. Applying the property, we subtract the exponents $\frac{7{5} - \frac{7}{5} = 0$. So, our expression simplifies to 303^0. Now, we apply another fundamental property of exponents: any non-zero number raised to the power of 0 is 1. Therefore, 303^0 equals 1. This completes the simplification process, and we have successfully rewritten the expression to its simplest form.

Final Answer

After meticulously applying the properties of exponents, we have successfully simplified the expression $ rac{3 imes 3{\frac{2}{5}}}{9{\frac{7}{10}}}$ to its final form. We began by expressing all bases in their prime factors, which allowed us to apply the power of a power property, the product of powers property, and finally, the quotient of powers property. Through these steps, we transformed the original expression into a much simpler form. The final step involved recognizing that any non-zero number raised to the power of 0 is 1. Therefore, the simplified form of the expression is:

3imes3259710=1\frac{3 imes 3^{\frac{2}{5}}}{9^{\frac{7}{10}}} = 1

This result not only provides the answer but also demonstrates the power and elegance of exponent properties in simplifying complex mathematical expressions. Understanding and applying these properties are crucial for further studies in mathematics and related fields. The ability to manipulate exponents efficiently is a foundational skill that opens doors to more advanced mathematical concepts and problem-solving techniques.

Conclusion

In conclusion, mastering the properties of exponents is not just about memorizing rules; it's about developing a deep understanding of how these rules can be applied to simplify complex expressions. Throughout this article, we have meticulously broken down the process of rewriting the expression $\frac{3 imes 3{\frac{2}{5}}}{9{\frac{7}{10}}}$, demonstrating each step with clarity and precision. We began by understanding the basics of exponents, delved into key properties such as the product of powers, quotient of powers, and power of a power, and then applied these properties systematically to simplify the expression. The final result, 1, showcases the power of these properties in transforming a seemingly complex expression into a simple, elegant form.

This exercise underscores the importance of a step-by-step approach in mathematics. By breaking down a complex problem into smaller, manageable parts, we can apply the appropriate rules and properties with confidence. Moreover, this process highlights the interconnectedness of mathematical concepts. The properties of exponents are not isolated rules; they are integral to many areas of mathematics, including algebra, calculus, and beyond. A strong grasp of these concepts is essential for anyone pursuing further studies in these fields. Therefore, continuous practice and application of these properties are key to achieving mastery and unlocking the door to more advanced mathematical problem-solving.