Evaluating Exponential Expressions A Step-by-Step Guide

This comprehensive guide dives deep into the evaluation of expressions involving exponents, covering a range of examples from basic fractional exponents to more complex combinations of powers and division. Mastering these concepts is crucial for success in algebra and beyond. Let's embark on this mathematical journey!

a. Evaluating (-7/10)^3

When evaluating expressions with exponents, understanding the impact of negative signs and fractional bases is paramount. In this first example, (-7/10)^3, we are dealing with a negative fraction raised to the power of 3. This means we need to multiply the fraction -7/10 by itself three times. The key here is to remember that a negative number raised to an odd power will result in a negative number, while a negative number raised to an even power will result in a positive number. So, let's break it down:

(-7/10)^3 = (-7/10) * (-7/10) * (-7/10)

First, let's multiply the first two fractions: (-7/10) * (-7/10). When multiplying fractions, we multiply the numerators together and the denominators together. Also, a negative times a negative is a positive. So, we have:

(-7 * -7) / (10 * 10) = 49/100

Now, we multiply this result by the remaining fraction, -7/10:

(49/100) * (-7/10) = (49 * -7) / (100 * 10)

Multiplying the numerators gives us 49 * -7 = -343, and multiplying the denominators gives us 100 * 10 = 1000. Therefore, the final result is:

-343/1000

This fraction cannot be simplified further, so our final answer is -343/1000. This example highlights the importance of paying close attention to signs and correctly applying the rules of fraction multiplication when working with exponents. The result, a negative fraction, underscores the principle that a negative base raised to an odd power yields a negative result. By meticulously following each step, we arrive at the correct evaluation, reinforcing the fundamental principles of exponent manipulation.

b. Simplifying (-8)^3 × (-5)^3

This problem, (-8)^3 × (-5)^3, provides an excellent opportunity to utilize the properties of exponents to simplify calculations. A key property to remember is that (a * b)^n = a^n * b^n. However, our problem is in the reverse format: a^n * b^n. We can apply the property in reverse as well, rewriting the expression as (a * b)^n. In this case, we have (-8)^3 × (-5)^3, which can be rewritten as:

(-8 * -5)^3

First, let's multiply the bases inside the parentheses: -8 * -5. A negative times a negative is a positive, so we get:

40

Now, we have:

(40)^3

This means we need to multiply 40 by itself three times:

40 * 40 * 40

Let's start by multiplying the first two 40s:

40 * 40 = 1600

Now, we multiply this result by the remaining 40:

1600 * 40 = 64000

Therefore, the final answer is 64000. This approach dramatically simplifies the calculation. Instead of computing (-8)^3 and (-5)^3 separately and then multiplying the results (which would involve larger numbers and a higher chance of making a mistake), we combined the bases first. This strategy underscores the power and efficiency of exponent properties in simplifying complex expressions. By recognizing and applying the appropriate property, we transformed a potentially cumbersome calculation into a straightforward one, highlighting the elegance and utility of mathematical principles.

c. Understanding Zero Exponents: 815^0 × 99^0

This part of the problem, 815^0 × 99^0, introduces a fundamental concept in exponents: the zero exponent rule. This rule states that any non-zero number raised to the power of 0 is equal to 1. This rule might seem counterintuitive at first, but it is a cornerstone of exponent arithmetic and ensures consistency within the system of exponents. Applying this rule, we can immediately simplify our expression.

Let's consider each term separately. First, we have 815^0. According to the zero exponent rule, this is equal to 1:

815^0 = 1

Similarly, we have 99^0. Applying the same rule, this is also equal to 1:

99^0 = 1

Now, we substitute these values back into our original expression:

1 × 1

The product of 1 and 1 is simply 1. Therefore, the final answer is:

1

This example beautifully illustrates the power and simplicity of the zero exponent rule. It transforms what might appear to be a complex calculation involving large numbers into a trivial one. The zero exponent rule is not just a mathematical curiosity; it is a critical component of the broader system of exponents. It plays a vital role in various mathematical contexts, including polynomial arithmetic, scientific notation, and more. Understanding and applying this rule correctly is essential for anyone working with exponential expressions.

d. Evaluating 65(47^0 + 33^0)

This expression, 65(47^0 + 33^0), combines the zero exponent rule with the order of operations, providing an excellent exercise in applying mathematical principles. As we learned in the previous example, any non-zero number raised to the power of 0 equals 1. This fact is crucial for simplifying this expression. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates that we must first evaluate the expressions inside the parentheses and then perform any exponentiation before moving on to multiplication.

Inside the parentheses, we have two terms, each raised to the power of 0: 47^0 and 33^0. Applying the zero exponent rule, we know that:

47^0 = 1

and

33^0 = 1

Now, we can substitute these values back into the expression inside the parentheses:

(1 + 1)

Adding these together, we get:

2

Now our expression simplifies to:

65 * 2

This is a simple multiplication problem. Multiplying 65 by 2, we get:

130

Therefore, the final answer is 130. This example demonstrates the importance of following the order of operations and correctly applying the zero exponent rule. By breaking the problem down into smaller steps and systematically addressing each component, we can arrive at the correct solution. This approach is crucial for tackling more complex mathematical problems and ensures accuracy in our calculations. The combination of exponent rules and order of operations is a fundamental aspect of mathematical literacy, and mastering these concepts is essential for further studies in mathematics and related fields.

e. Applying Exponent Rules: (9/11) × (9/11)^5 × (9/11)^-3

This problem, (9/11) × (9/11)^5 × (9/11)^-3, is a fantastic illustration of how exponent rules can simplify expressions with the same base. The core rule we'll use here is the product of powers rule, which states that when multiplying exponents with the same base, you add the powers: a^m * a^n = a^(m+n). The key to solving this problem efficiently is recognizing that all three terms have the same base, 9/11, which allows us to directly apply this rule.

The first term, (9/11), can be thought of as (9/11)^1, as any number raised to the power of 1 is itself. So, we can rewrite the expression as:

(9/11)^1 × (9/11)^5 × (9/11)^-3

Now, we can apply the product of powers rule by adding the exponents:

(9/11)^(1 + 5 + (-3))

Adding the exponents, we get:

1 + 5 + (-3) = 3

So our expression simplifies to:

(9/11)^3

This means we need to multiply 9/11 by itself three times:

(9/11) * (9/11) * (9/11)

When multiplying fractions, we multiply the numerators together and the denominators together:

(9 * 9 * 9) / (11 * 11 * 11)

Calculating the numerator, we have:

9 * 9 * 9 = 729

Calculating the denominator, we have:

11 * 11 * 11 = 1331

Therefore, the final answer is:

729/1331

This fraction cannot be simplified further, so our solution is complete. This example effectively demonstrates the power of the product of powers rule in simplifying expressions. By recognizing the common base and applying the rule, we transformed a potentially cumbersome calculation into a straightforward one. This skill is essential for tackling more complex algebraic expressions and is a cornerstone of mathematical fluency.

f. Dividing Exponents with the Same Base: (23/70)^3 ÷ (23/70)^2

This problem, (23/70)^3 ÷ (23/70)^2, focuses on division of exponents with the same base. The rule that governs this situation is the quotient of powers rule, which states that when dividing exponents with the same base, you subtract the exponents: a^m / a^n = a^(m-n). This rule is a direct consequence of the properties of exponents and is invaluable for simplifying expressions.

In this case, our base is 23/70, and we are dividing (23/70)^3 by (23/70)^2. Applying the quotient of powers rule, we subtract the exponents:

(23/70)^(3 - 2)

Subtracting the exponents, we get:

3 - 2 = 1

So our expression simplifies to:

(23/70)^1

Any number raised to the power of 1 is simply itself. Therefore, the final answer is:

23/70

This fraction is already in its simplest form, so our solution is complete. This example elegantly demonstrates the quotient of powers rule. By recognizing the common base and applying the rule, we bypassed the need to calculate the individual powers, resulting in a much simpler solution. This skill is crucial for handling algebraic fractions and complex exponential expressions. Mastering the quotient of powers rule allows for efficient simplification and accurate calculations, which are essential for success in advanced mathematical topics.

g. Working with Negative Exponents in Division: (15/47)^-3 ÷ (15/47)^-2

This example, (15/47)^-3 ÷ (15/47)^-2, delves into the division of exponents with the same base, but this time with negative exponents. This scenario combines the quotient of powers rule with the concept of negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. For example, a^-n = 1/a^n. We'll apply both this understanding and the quotient of powers rule to solve the problem.

The quotient of powers rule, as we saw in the previous example, states that when dividing exponents with the same base, you subtract the exponents: a^m / a^n = a^(m-n). Applying this rule to our expression, we have:

(15/47)^(-3 - (-2))

The key here is to carefully handle the subtraction of the negative exponents. Subtracting a negative number is the same as adding its positive counterpart. So, we have:

-3 - (-2) = -3 + 2 = -1

Our expression now simplifies to:

(15/47)^-1

Now, we apply the rule for negative exponents. A number raised to the power of -1 is the reciprocal of that number. Therefore:

(15/47)^-1 = 47/15

This is our final answer. We have successfully navigated the negative exponents and the division, arriving at a simplified fraction. This example underscores the importance of understanding and correctly applying both the quotient of powers rule and the properties of negative exponents. These concepts are fundamental to algebraic manipulation and are essential tools for solving a wide range of mathematical problems. The ability to work confidently with negative exponents is a hallmark of strong algebraic skills.

h. Combining Exponent Rules: (3^2 × 243^-1) / (81^-1 × 729)

This complex expression, (3^2 × 243^-1) / (81^-1 × 729), is a comprehensive exercise in applying multiple exponent rules. To effectively solve this problem, we'll need to utilize the product of powers rule, the quotient of powers rule, the negative exponent rule, and the ability to express numbers as powers of a common base. The common base in this case is 3, as 243, 81, and 729 can all be expressed as powers of 3. This is the key to simplifying the expression.

First, let's express each term as a power of 3:

• 3^2 remains as 3^2
• 243 = 3^5, so 243^-1 = (3⁵⁾-1 = 3^-5 (using the power of a power rule: (a^m)n = a^(m*n))
• 81 = 3^4, so 81^-1 = (3⁴⁾-1 = 3^-4
• 729 = 3^6

Now, we substitute these values back into our original expression:

(3^2 × 3^-5) / (3^-4 × 3^6)

Next, we apply the product of powers rule in both the numerator and the denominator. In the numerator, we have:

3^2 × 3^-5 = 3^(2 + (-5)) = 3^-3

In the denominator, we have:

3^-4 × 3^6 = 3^(-4 + 6) = 3^2

Our expression now looks like this:

3^-3 / 3^2

Now, we apply the quotient of powers rule:

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

Finally, we use the negative exponent rule to express the result with a positive exponent:

3^-5 = 1 / 3^5

We need to calculate 3^5, which is 3 * 3 * 3 * 3 * 3 = 243. Therefore, the final answer is:

1/243

This problem exemplifies how a combination of exponent rules can be used to simplify a complex expression. By strategically applying the rules and expressing the terms with a common base, we transformed a seemingly daunting problem into a manageable one. This level of algebraic manipulation is crucial for advanced mathematical studies and demonstrates a strong command of exponent properties.

i. Simplifying Expressions with Fractional Bases: (6/42)^3 × (7/9)^2

This problem, (6/42)^3 × (7/9)^2, involves simplifying expressions with fractional bases raised to powers. The key to tackling this problem is to first simplify the fractions within the parentheses and then apply the exponent rules. Simplifying the fractions at the beginning makes the subsequent calculations much easier and reduces the risk of errors.

Let's start by simplifying the fraction 6/42. Both 6 and 42 are divisible by 6. Dividing both the numerator and the denominator by 6, we get:

6/42 = 1/7

So, our expression becomes:

(1/7)^3 × (7/9)^2

Now, we need to raise each fraction to its respective power. Let's start with (1/7)^3. This means we multiply 1/7 by itself three times:

(1/7)^3 = (1/7) * (1/7) * (1/7) = 1/343

Next, we need to calculate (7/9)^2. This means we multiply 7/9 by itself:

(7/9)^2 = (7/9) * (7/9) = 49/81

Now, our expression looks like this:

(1/343) × (49/81)

To multiply fractions, we multiply the numerators together and the denominators together:

(1 * 49) / (343 * 81) = 49 / 27783

Now, we need to see if we can simplify this fraction. Both 49 and 343 are divisible by 49. In fact, 343 = 49 * 7. So, we can simplify the fraction as follows:

49 / 27783 = 49 / (49 * 7 * 81) = 1 / (7 * 81) = 1/567

Therefore, the final answer is:

1/567

This problem demonstrates the importance of simplifying fractions before applying exponents and then simplifying the resulting fraction. By breaking down the problem into smaller, manageable steps, we were able to arrive at the solution efficiently. This approach is crucial for working with fractional exponents and complex algebraic expressions.

j. Applying Exponent Rules to Fractions: (9/16)^2

This final problem, (9/16)^2, provides a straightforward application of exponent rules to fractions. When a fraction is raised to a power, it means that both the numerator and the denominator are raised to that power. This is a fundamental rule for handling exponents with fractional bases. Applying this rule correctly is key to solving the problem efficiently.

In this case, we have (9/16)^2. This means we need to raise both the numerator, 9, and the denominator, 16, to the power of 2:

(9/16)^2 = 9^2 / 16^2

Now, we calculate each power separately. First, let's calculate 9^2:

9^2 = 9 * 9 = 81

Next, let's calculate 16^2:

16^2 = 16 * 16 = 256

So, our expression becomes:

81 / 256

Now, we need to determine if this fraction can be simplified. The factors of 81 are 1, 3, 9, 27, and 81. The factors of 256 are powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256). Since they share no common factors other than 1, the fraction is already in its simplest form. Therefore, the final answer is:

81/256

This example clearly illustrates how to raise a fraction to a power. By applying the rule of raising both the numerator and denominator to the power, we simplified the problem into calculating individual powers. This is a crucial skill for working with fractional exponents and is a building block for more complex algebraic manipulations. The ability to efficiently and accurately handle fractional exponents is essential for success in various mathematical contexts.

In conclusion, evaluating expressions involving exponents requires a solid understanding of exponent rules, careful attention to signs, and the ability to break down complex problems into manageable steps. From understanding zero exponents to applying the product and quotient of powers rules, these examples provide a comprehensive foundation for mastering exponent manipulation. By practicing these techniques, you'll build confidence and proficiency in algebra and beyond. Mastering exponents is not just about memorizing rules; it's about understanding the underlying principles and applying them strategically to solve problems. This understanding will empower you to tackle more advanced mathematical concepts with ease and confidence. Remember to always simplify where possible, double-check your work, and embrace the challenge of problem-solving. With consistent effort and a solid grasp of the fundamentals, you can excel in your mathematical journey.