Solving Mathematical Expressions A Step-by-Step Guide
In the realm of mathematics, encountering complex expressions is a common challenge. These expressions often involve a combination of various operations, exponents, and fractions, demanding a systematic approach to arrive at the correct solution. This comprehensive guide aims to demystify the process of solving mathematical expressions, focusing on the specific example of (−1/8)³ × (-1)¹⁰⁵ × (8/5)². By dissecting each step and providing clear explanations, we will equip you with the skills and knowledge necessary to tackle similar problems with confidence.
Understanding the Order of Operations
Before diving into the specifics of our example, it's crucial to grasp the fundamental principle governing mathematical expressions: the order of operations. This principle dictates the sequence in which operations must be performed to ensure consistent and accurate results. The widely recognized acronym PEMDAS serves as a helpful mnemonic for remembering this order:
- Parentheses (or Brackets)
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Adhering to PEMDAS is paramount in solving mathematical expressions. Failing to do so can lead to incorrect answers and a misunderstanding of the underlying mathematical principles. With this foundation in place, we can now embark on the step-by-step solution of our target expression.
Step 1: Evaluating Exponents
The first step in simplifying the expression (−1/8)³ × (-1)¹⁰⁵ × (8/5)² is to evaluate the exponents. Exponents indicate the number of times a base is multiplied by itself. Let's break down each term:
Term 1: (−1/8)³
This term signifies (-1/8) multiplied by itself three times:
(−1/8)³ = (-1/8) × (-1/8) × (-1/8)
When multiplying fractions, we multiply the numerators (top numbers) and the denominators (bottom numbers) separately. Additionally, we must consider the signs. The product of three negative numbers is negative.
(-1 × -1 × -1) / (8 × 8 × 8) = -1 / 512
Therefore, (−1/8)³ simplifies to -1/512.
Term 2: (-1)¹⁰⁵
This term represents -1 raised to the power of 105. A crucial concept to remember is that a negative number raised to an odd power results in a negative number, while a negative number raised to an even power results in a positive number. Since 105 is an odd number, (-1)¹⁰⁵ is:
(-1)¹⁰⁵ = -1
Term 3: (8/5)²
This term indicates (8/5) multiplied by itself:
(8/5)² = (8/5) × (8/5)
Multiplying the numerators and denominators, we get:
(8 × 8) / (5 × 5) = 64 / 25
Thus, (8/5)² simplifies to 64/25.
Now that we have evaluated the exponents, our expression becomes:
-1/512 × -1 × 64/25
Step 2: Performing Multiplication
The next step, according to PEMDAS, is to perform the multiplication. We have three terms to multiply:
-1/512 × -1 × 64/25
When multiplying fractions, we multiply the numerators and the denominators. We also need to consider the signs. The product of two negative numbers is positive.
First, let's multiply -1/512 by -1:
(-1/512) × (-1) = 1/512
Now, we multiply the result by 64/25:
(1/512) × (64/25) = (1 × 64) / (512 × 25)
This simplifies to:
64 / 12800
Step 3: Simplifying the Fraction
The final step is to simplify the fraction 64/12800 to its lowest terms. Both the numerator and the denominator are divisible by 64. Dividing both by 64, we get:
64 ÷ 64 / 12800 ÷ 64 = 1 / 200
Therefore, the simplified result of the expression (−1/8)³ × (-1)¹⁰⁵ × (8/5)² is 1/200.
Alternative Approach: Prime Factorization
An alternative method to simplify fractions, particularly useful with larger numbers, is prime factorization. Prime factorization involves breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number. Let's apply this to our fraction 64/12800.
Prime Factorization of 64:
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶
Prime Factorization of 12800:
12800 = 2 × 6400 6400 = 2 × 3200 3200 = 2 × 1600 1600 = 2 × 800 800 = 2 × 400 400 = 2 × 200 200 = 2 × 100 100 = 2 × 50 50 = 2 × 25 25 = 5 × 5
So, 12800 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 2⁸ × 5²
Now we can rewrite the fraction 64/12800 using prime factors:
64/12800 = 2⁶ / (2⁸ × 5²)
To simplify, we cancel out common factors. We have 2⁶ in the numerator and 2⁸ in the denominator. Canceling out 2⁶ from both, we are left with:
1 / (2² × 5²)
Which simplifies to:
1 / (4 × 25) = 1 / 100
Identifying the Error and Correcting the Calculation
Upon reviewing the prime factorization method, an error was identified in the initial simplification. Let's pinpoint the mistake and correct the calculation.
We correctly determined the prime factorizations:
- 64 = 2⁶
- 12800 = 2⁸ × 5²
The fraction was then expressed as:
64/12800 = 2⁶ / (2⁸ × 5²)
The error occurred in the simplification after canceling out the common factors. When we cancel out 2⁶ from both the numerator and the denominator, we are left with:
1 / (2^(8-6) × 5²)
Which simplifies to:
1 / (2² × 5²)
This is where the error was made. The correct simplification should be:
1 / (4 × 25) = 1 / 100
However, this result still contradicts our initial solution of 1/200. Let's re-examine the original calculation in Step 2 to identify any potential errors.
Re-evaluating Step 2: Multiplication
In Step 2, we performed the multiplication:
-1/512 × -1 × 64/25
We correctly multiplied -1/512 by -1 to get 1/512. Then, we multiplied 1/512 by 64/25:
(1/512) × (64/25) = (1 × 64) / (512 × 25) = 64 / 12800
The error lies in the simplification of 64/12800. We previously stated that dividing both numerator and denominator by 64 gives us 1/200. Let's verify this:
64 ÷ 64 = 1 12800 ÷ 64 = 200
So, 64/12800 does indeed simplify to 1/200.
Reconciliation and Final Answer
We have identified an error in the prime factorization method, which led to an incorrect result of 1/100. The correct simplification of the fraction 64/12800, as determined in Step 3, is 1/200. This aligns with our initial calculation.
Therefore, the final answer to the expression (−1/8)³ × (-1)¹⁰⁵ × (8/5)² is 1/200.
Conclusion
Solving complex mathematical expressions requires a systematic approach and a thorough understanding of the order of operations. By meticulously evaluating exponents, performing multiplication, and simplifying fractions, we successfully navigated the expression (−1/8)³ × (-1)¹⁰⁵ × (8/5)² and arrived at the solution of 1/200. This step-by-step guide not only provides the answer but also equips you with the knowledge and skills to tackle similar challenges in the future. Remember, practice is key to mastering mathematical concepts, so continue to challenge yourself with diverse problems and solidify your understanding.
