Simplifying Exponential Expressions Using Laws Of Exponents

In mathematics, laws of exponents provide a set of rules for simplifying expressions involving exponents. These laws are essential tools for manipulating and solving equations with powers and roots. When we work with exponential expressions, we often encounter situations where we need to simplify them to obtain the most concise and manageable form. In this article, we'll explore a specific example of simplifying an exponential expression using the laws of exponents. This comprehensive guide will provide you with a deep understanding of how to effectively apply these rules, ensuring you can confidently tackle similar problems in the future. Grasping the laws of exponents is crucial for success in algebra, calculus, and other advanced mathematical fields. By mastering these concepts, you'll enhance your ability to solve complex problems and gain a solid foundation in mathematics.

The problem we will address involves simplifying a fraction where both the numerator and the denominator are exponential terms with the same base. This type of problem frequently appears in various mathematical contexts, including algebraic manipulations, calculus problems, and scientific calculations. To simplify such expressions, we rely on one of the fundamental laws of exponents, which allows us to divide exponential terms with the same base by subtracting their exponents. This rule is not only efficient but also crucial for streamlining complex expressions into a more manageable format. By understanding and applying this law, you can significantly reduce the complexity of mathematical problems and arrive at accurate solutions more easily. Our step-by-step approach in this article will break down the simplification process, making it clear and straightforward for learners of all levels.

Simplify the following expression:

$$\frac{6^{\frac{7}{6}}}{6^{\frac{5}{6}}} = \square$$

(Simplify your answer. Type exponential notation with a rational exponent.)

This problem requires us to simplify a fraction involving exponential terms with the same base. The base in this case is 6, and the exponents are rational numbers, specifically ${\frac{7}{6}}$ and ${\frac{5}{6}}$. To solve this, we will apply one of the key laws of exponents, which deals with the division of exponential terms sharing a common base. This law allows us to simplify the expression efficiently and accurately. The presence of rational exponents adds a layer of complexity, but by understanding the underlying principles, we can handle such expressions with confidence. Mastering the simplification of exponential expressions with rational exponents is a fundamental skill in algebra and higher mathematics, as it appears frequently in various contexts, including calculus, differential equations, and mathematical modeling. By working through this problem, you will not only reinforce your understanding of exponent laws but also enhance your problem-solving abilities in mathematics.

The laws of exponents are a set of rules that allow us to simplify expressions involving powers. These laws are fundamental in algebra and calculus. One of the key laws we'll use here is the quotient of powers rule. The quotient of powers rule is a cornerstone of exponential arithmetic, allowing us to simplify expressions involving division of powers with the same base. This rule states that when dividing two exponential expressions with the same base, you subtract the exponents. This law is expressed mathematically as:

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

where a is the base, and m and n are the exponents. This rule is essential for simplifying complex expressions and is widely used in various mathematical and scientific contexts. By mastering the quotient of powers rule, you can efficiently manipulate exponential expressions, making it easier to solve equations, analyze functions, and perform calculations in fields such as physics, engineering, and computer science. Understanding the conditions under which this rule applies, such as the base not being zero, is also crucial for accurate application. The quotient of powers rule not only simplifies computations but also provides a deeper insight into the behavior of exponential functions, which are ubiquitous in both theoretical and applied mathematics.

In this rule:

• a is the base.
• m and n are the exponents.

This law is applicable when dividing exponential terms with the same base. Understanding this rule is crucial for simplifying expressions like the one we have in the problem statement. The quotient of powers rule is just one of several laws of exponents, but it is particularly useful in situations where we have fractions involving exponential terms. By applying this rule, we can reduce complex expressions to simpler forms, making them easier to work with. For example, consider the expression ${\frac{x^5}{x^2}}$. Using the quotient of powers rule, we subtract the exponents to get ${x^{5-2} = x^3}$. This simplification not only makes the expression more manageable but also reveals the underlying mathematical relationship between the numerator and the denominator. Mastering the quotient of powers rule is, therefore, a vital step in developing strong algebraic skills.

To simplify the given expression:

$$\frac{6^{\frac{7}{6}}}{6^{\frac{5}{6}}}$$

We apply the quotient of powers rule, which states that when dividing exponential terms with the same base, we subtract the exponents. In this case, the base is 6, and the exponents are ${\frac{7}{6}}$ and ${\frac{5}{6}}$. Applying the rule, we subtract the exponent in the denominator from the exponent in the numerator:

$$6^{\frac{7}{6} - \frac{5}{6}}$$

This step is crucial because it transforms a division problem into a subtraction problem in the exponent, which is often easier to handle. The quotient of powers rule is a fundamental tool for simplifying exponential expressions, and its application here allows us to reduce the given fraction to a more manageable form. By subtracting the exponents, we are essentially finding the net effect of the division on the power of the base. This technique is not only useful for numerical calculations but also for algebraic manipulations where exponential terms need to be combined or simplified. Mastering this step is essential for further simplifying the expression and arriving at the final answer. The ability to correctly apply the quotient of powers rule is a key skill in algebra and higher mathematics.

Now, we perform the subtraction in the exponent. The subtraction ${\frac{7}{6} - \frac{5}{6}}$ involves fractions with a common denominator, which simplifies the process:

$$\frac{7}{6} - \frac{5}{6} = \frac{7 - 5}{6} = \frac{2}{6}$$

This subtraction is straightforward because the fractions have the same denominator. When fractions have a common denominator, you simply subtract the numerators and keep the denominator the same. This basic arithmetic operation is a fundamental skill in mathematics and is essential for working with rational exponents. The result of the subtraction, ${\frac{2}{6}}$, represents the simplified exponent that we will use in the next step. Accurate subtraction of fractions is crucial for obtaining the correct final answer in this type of problem. A solid understanding of fraction arithmetic is, therefore, a prerequisite for mastering exponential operations involving rational exponents. This step-by-step approach ensures that each component of the simplification process is clear and easy to follow.

We can further simplify the fraction ${\frac{2}{6}}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$$\frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3}$$

Simplifying fractions is a crucial step in mathematical problem-solving, as it reduces the expression to its simplest form, making it easier to understand and work with. In this case, dividing both the numerator and the denominator of ${\frac{2}{6}}$ by 2 gives us the simplified fraction ${\frac{1}{3}}$. This simplification is important because it presents the exponent in its most concise form, which is often preferred in mathematical notation. The ability to recognize and perform fraction simplification is a fundamental skill that enhances mathematical fluency. Simplified fractions not only make subsequent calculations easier but also provide a clearer representation of the numerical relationship. This step ensures that our final answer will be in its most reduced and understandable form.

Now, substitute the simplified exponent back into the expression:

$$6^{\frac{2}{6}} = 6^{\frac{1}{3}}$$

This step involves replacing the original complex exponent with its simplified form, which we obtained in the previous steps. Substituting the simplified exponent ${\frac{1}{3}}$ back into the expression makes the expression cleaner and easier to interpret. This process demonstrates the importance of simplifying intermediate results to make the final calculation more manageable. The ability to substitute values and expressions correctly is a fundamental skill in algebra and mathematics in general. By performing this substitution, we are one step closer to the final, simplified answer. This step also highlights the logical flow of the simplification process, where each step builds upon the previous one to gradually reduce the complexity of the expression.

Therefore, the simplified form of the expression is:

$$6^{\frac{1}{3}}$$

This is the final simplified form of the given expression. The result, ${6^{\frac{1}{3}}}$ represents the cube root of 6. This final step demonstrates the power of the laws of exponents in simplifying complex expressions into more manageable forms. The ability to arrive at a concise and simplified answer is a testament to a strong understanding of exponential rules and algebraic manipulation. This result not only provides the solution to the problem but also highlights the connection between exponents and roots, as a fractional exponent is equivalent to a radical expression. The simplified form ${6^{\frac{1}{3}}}$ is both mathematically accurate and easily interpretable, making it the preferred way to express the result. Mastering this type of simplification is essential for success in various areas of mathematics and its applications.

In summary, we simplified the expression ${\frac{6^{\frac{7}{6}}}{6^{\frac{5}{6}}}}$ using the quotient of powers rule. By subtracting the exponents and simplifying the resulting fraction, we arrived at the final answer ${6^{\frac{1}{3}}}$ This exercise demonstrates the importance and effectiveness of the laws of exponents in simplifying mathematical expressions. The step-by-step approach we followed highlights the logical progression required to solve such problems, emphasizing the need for accurate application of rules and simplification techniques. Mastering these skills is crucial for success in algebra and higher mathematics. The ability to simplify expressions efficiently and accurately not only makes problem-solving easier but also provides a deeper understanding of the underlying mathematical concepts. This example serves as a valuable lesson in how to approach and solve exponential problems, reinforcing the fundamental principles of algebra and paving the way for more advanced mathematical studies.

The laws of exponents are powerful tools that simplify complex mathematical expressions. Understanding and applying these laws is essential for success in mathematics and related fields. This article has provided a detailed walkthrough of how to use the quotient of powers rule, a fundamental concept in algebra. By mastering these techniques, you can confidently tackle more challenging problems and deepen your mathematical understanding. The skills acquired through this exercise are transferable to various mathematical contexts, including calculus, differential equations, and mathematical modeling. Continued practice and application of these concepts will solidify your understanding and enhance your problem-solving abilities. The laws of exponents are not just abstract rules; they are practical tools that facilitate mathematical reasoning and computation, making complex calculations more manageable and accessible. By embracing these principles, you can unlock a greater appreciation for the elegance and efficiency of mathematical methods.