Expressing Numbers With Positive Exponents Using Laws Of Exponents

In mathematics, exponents play a crucial role in simplifying expressions and representing numbers in a concise manner. When dealing with exponents, understanding and applying the laws of exponents is essential. This article will delve into the laws of exponents and demonstrate how to express various expressions with positive exponents. We will explore several examples, providing step-by-step solutions and explanations to enhance your understanding of this fundamental concept. Mastering the laws of exponents will not only aid in simplifying mathematical problems but also lay a strong foundation for more advanced topics in algebra and calculus. Let's embark on this journey to unravel the intricacies of exponents and their applications.

(i) (2/3)^(-5)

The first expression we will tackle is (2/3)^(-5). This involves a fraction raised to a negative exponent. To express this with a positive exponent, we need to apply the rule that states a^(-n) = 1/a^(n). This rule essentially tells us that a number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent. Applying this rule to our expression, we get:

(2/3)^(-5) = 1 / (2/3)^(5)

Now, we have a fraction raised to a positive exponent in the denominator. To further simplify, we can rewrite the denominator as the reciprocal raised to the power of 5. The reciprocal of 2/3 is 3/2. So, we can rewrite the expression as:

1 / (2/3)^(5) = (3/2)^(5)

Thus, the expression (2/3)^(-5), when expressed with a positive exponent, becomes (3/2)^(5). This demonstrates the fundamental principle of converting negative exponents to positive exponents by taking the reciprocal of the base. The positive exponent now indicates that we need to multiply the fraction 3/2 by itself five times. This transformation is crucial in simplifying complex expressions and making them easier to evaluate.

To further illustrate, let's break down the calculation of (3/2)^(5):

(3/2)^(5) = (3/2) * (3/2) * (3/2) * (3/2) * (3/2) = 243/32

This result shows that (3/2)^(5) is equivalent to the fraction 243/32. The process of converting the negative exponent to a positive exponent not only simplifies the form of the expression but also allows for easier computation and understanding of its value. This fundamental concept is widely used in various mathematical applications, including scientific calculations and engineering problems. Therefore, mastering this rule is essential for anyone working with exponential expressions.

(ii) (2^(-4))2

The next expression we will consider is (2^(-4))(2). This involves an exponent raised to another exponent, which necessitates the use of the power of a power rule. The rule states that (a^(m))(n) = a^(mn)*. In other words, when you raise a power to another power, you multiply the exponents. Applying this rule to our expression, we get:

(2^(-4))(2) = 2^(-4 * 2) = 2^(-8)

Now, we have a base raised to a negative exponent. Similar to the previous example, we need to convert this to a positive exponent. Using the rule a^(-n) = 1/a^(n), we can rewrite the expression as:

2^(-8) = 1 / 2^(8)

Thus, the expression (2^(-4))(2), when expressed with a positive exponent, becomes 1 / 2^(8). This transformation demonstrates the application of two key rules of exponents: the power of a power rule and the rule for negative exponents. The positive exponent now indicates that we need to calculate 2 raised to the power of 8 and then take its reciprocal.

To further elaborate, let's calculate 2^(8):

2^(8) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256

Therefore, the expression becomes:

1 / 2^(8) = 1 / 256

This result shows that (2^(-4))(2) simplifies to 1/256. The step-by-step application of the exponent rules allows us to convert a complex exponential expression into a simple fraction. This process is fundamental in simplifying algebraic expressions and solving equations involving exponents. Understanding these rules is crucial for anyone working with mathematical and scientific computations, as they provide a systematic way to handle exponential operations.

(iii) 4^3 × 4^(-5)

Our third example is 4^3 × 4^(-5). This expression involves the multiplication of two terms with the same base but different exponents. To simplify this, we use the product of powers rule, which states that a^(m) × a^(n) = a^(m+n). Applying this rule to our expression, we get:

4^3 × 4^(-5) = 4^(3 + (-5)) = 4^(-2)

Now, we have a base raised to a negative exponent. To express this with a positive exponent, we use the rule a^(-n) = 1/a^(n):

4^(-2) = 1 / 4^(2)

Thus, the expression 4^3 × 4^(-5), when expressed with a positive exponent, becomes 1 / 4^(2). This illustrates the use of the product of powers rule and the conversion of a negative exponent to a positive exponent. The positive exponent now indicates that we need to calculate 4 squared and then take its reciprocal.

To further clarify, let's calculate 4^(2):

4^(2) = 4 * 4 = 16

Therefore, the expression becomes:

1 / 4^(2) = 1 / 16

This result shows that 4^3 × 4^(-5) simplifies to 1/16. The combination of the product of powers rule and the negative exponent rule allows us to efficiently simplify expressions involving exponents. This process is essential in various mathematical contexts, including algebraic manipulations and problem-solving. Mastering these rules provides a solid foundation for more advanced topics in mathematics and science, where exponents are frequently encountered.

(iv) [(3/2)^(-2)]3

Now, let's consider the expression [(3/2)^(-2)]3. This involves a fraction raised to a negative exponent, all of which is then raised to another exponent. We will again use the power of a power rule, which states that (a^(m))(n) = a^(mn)*. Applying this rule to our expression, we get:

[(3/2)^(-2)]3 = (3/2)^(-2 * 3) = (3/2)^(-6)

We now have a fraction raised to a negative exponent. To express this with a positive exponent, we use the rule a^(-n) = 1/a^(n). However, a more direct approach for fractions is to take the reciprocal of the base and change the sign of the exponent. So, we take the reciprocal of 3/2, which is 2/3, and change the exponent from -6 to 6:

(3/2)^(-6) = (2/3)^(6)

Thus, the expression [(3/2)^(-2)]3, when expressed with a positive exponent, becomes (2/3)^(6). This demonstrates the combined application of the power of a power rule and the reciprocal property for negative exponents. The positive exponent now indicates that we need to multiply the fraction 2/3 by itself six times.

To illustrate further, let's calculate (2/3)^(6):

(2/3)^(6) = (2/3) * (2/3) * (2/3) * (2/3) * (2/3) * (2/3) = 64/729

This result shows that [(3/2)^(-2)]3 simplifies to 64/729. The ability to apply these exponent rules efficiently is crucial in simplifying complex expressions and solving mathematical problems. This understanding is particularly valuable in fields that involve complex calculations, such as physics and engineering, where exponential expressions are common.

(v) 2^(-3) × (-7)^(-3)

Next, let's address the expression 2^(-3) × (-7)^(-3). This expression involves the multiplication of two terms with different bases but the same negative exponent. To handle this, we can use the rule that states a^(n) × b^(n) = (a × b)^(n). Applying this rule to our expression, we get:

2^(-3) × (-7)^(-3) = (2 × -7)^(-3) = (-14)^(-3)

Now, we have a negative number raised to a negative exponent. To express this with a positive exponent, we use the rule a^(-n) = 1/a^(n):

(-14)^(-3) = 1 / (-14)^(3)

Thus, the expression 2^(-3) × (-7)^(-3), when expressed with a positive exponent, becomes 1 / (-14)^(3). This illustrates the application of the rule for multiplying terms with the same exponent and the conversion of a negative exponent to a positive exponent. The positive exponent now indicates that we need to calculate -14 cubed and then take its reciprocal.

To further clarify, let's calculate (-14)^(3):

(-14)^(3) = -14 * -14 * -14 = -2744

Therefore, the expression becomes:

1 / (-14)^(3) = 1 / -2744 = -1/2744

This result shows that 2^(-3) × (-7)^(-3) simplifies to -1/2744. The ability to manipulate expressions with exponents, especially negative exponents, is a crucial skill in algebra and other branches of mathematics. This skill is particularly useful in simplifying complex expressions and making them easier to work with in problem-solving scenarios.

(vi) (2^5 + 2^8)

Finally, let's consider the expression (2^5 + 2^8). This expression involves the addition of two exponential terms. Unlike multiplication, there isn't a direct rule to simplify the addition of exponents with the same base. Instead, we need to calculate each term separately and then add them together.

First, let's calculate 2^5:

2^5 = 2 * 2 * 2 * 2 * 2 = 32

Next, let's calculate 2^8:

2^8 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 256

Now, we add these two results together:

2^5 + 2^8 = 32 + 256 = 288

Thus, the expression (2^5 + 2^8) simplifies to 288. This example illustrates that not all exponential expressions can be simplified using exponent rules directly. In cases involving addition or subtraction, it is often necessary to calculate the individual terms and then perform the operation. This understanding is crucial in handling a wide variety of mathematical problems involving exponents.

The expression (2^5 + 2^8) does not involve negative exponents, so there is no need to express it with a positive exponent. The final answer is simply the numerical value obtained by performing the addition. This type of problem highlights the importance of recognizing when to apply exponent rules and when to resort to direct calculation.

Express as a Rational Number with a Negative Exponent

Expressing a number as a rational number with a negative exponent involves understanding the relationship between fractions and negative exponents. A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. To express a rational number with a negative exponent, we need to rewrite the number in the form a^(-n), where a is the base and -n is the negative exponent. This process often involves recognizing the reciprocal relationship between a number and its exponent.

To illustrate, let's consider a few examples and break down the steps involved in expressing them with negative exponents.

Example 1: Express 1/16 as a rational number with a negative exponent.

First, we need to recognize that 16 is a power of 4. Specifically, 16 = 4^(2). Therefore, we can rewrite 1/16 as:

1/16 = 1/4^(2)

Now, we can use the rule a^(-n) = 1/a^(n) in reverse. This means that 1/a^(n) = a^(-n). Applying this rule, we get:

1/4^(2) = 4^(-2)

Thus, 1/16 can be expressed as 4^(-2), which is a rational number with a negative exponent. This example demonstrates how recognizing powers of a number can help in converting fractions to expressions with negative exponents.

Example 2: Express 1/27 as a rational number with a negative exponent.

In this case, we recognize that 27 is a power of 3. Specifically, 27 = 3^(3). Therefore, we can rewrite 1/27 as:

1/27 = 1/3^(3)

Using the rule 1/a^(n) = a^(-n), we can rewrite this as:

1/3^(3) = 3^(-3)

Thus, 1/27 can be expressed as 3^(-3), which is a rational number with a negative exponent. This further illustrates the process of converting fractions to expressions with negative exponents by identifying the base and its power.

Example 3: Express 256 as a rational number with a negative exponent.

This example is slightly different because we are starting with a whole number rather than a fraction. To express 256 with a negative exponent, we first need to express it as a fraction. We can do this by considering its reciprocal, 1/256.

We recognize that 256 is a power of 2. Specifically, 256 = 2^(8). Therefore, we can rewrite 1/256 as:

1/256 = 1/2^(8)

Using the rule 1/a^(n) = a^(-n), we can rewrite this as:

1/2^(8) = 2^(-8)

However, we started with 256, not 1/256. To express 256 with a negative exponent, we can rewrite it as:

256 = 1 / (1/256) = 1 / 2^(-8)

This form is technically correct, but it's more common to express it directly as a negative power of its reciprocal. So, another way to think about this is to find a base and exponent such that when the base is raised to the negative exponent, it equals 256. In this case, we can rewrite 256 as:

256 = (1/2)^(-8)

This might seem counterintuitive, but it's a valid representation. Alternatively, we can also consider other bases. For instance, 256 = 4^(4), so we could write:

256 = (1/4)^(-4)

Thus, 256 can be expressed as (1/2)^(-8) or (1/4)^(-4), both of which are rational numbers with negative exponents. This example demonstrates that there can be multiple ways to express a number with a negative exponent, depending on the base chosen.

In summary, expressing a number as a rational number with a negative exponent involves identifying the base and its power, understanding the reciprocal relationship, and applying the rule a^(-n) = 1/a^(n). By mastering these concepts, you can effectively manipulate and simplify expressions involving exponents, which is a crucial skill in mathematics and related fields.

In conclusion, mastering the laws of exponents is essential for simplifying mathematical expressions and solving complex problems. Throughout this article, we have explored various examples demonstrating how to express numbers with positive exponents and how to represent rational numbers with negative exponents. By understanding and applying rules such as the product of powers, power of a power, and the reciprocal property for negative exponents, you can efficiently manipulate exponential expressions. These skills are fundamental in mathematics, science, and engineering, providing a solid foundation for more advanced concepts and applications. Whether you are simplifying algebraic equations or tackling scientific calculations, a strong grasp of exponent rules will undoubtedly prove invaluable.