Calculating Values Of Rational Numbers Raised To Powers
This article provides a comprehensive guide to calculating the values of rational numbers raised to various powers. We will explore both positive and negative fractions raised to positive integer exponents, providing step-by-step solutions and explanations. This will enhance your understanding of exponents and fractions, crucial concepts in mathematics.
a. (3/4)^3
Exponents and Fractions: Understanding how to raise a fraction to a power is a fundamental concept in mathematics. In this case, we need to calculate (3/4)^3, which means multiplying the fraction 3/4 by itself three times. This calculation involves both the numerator and the denominator, and a clear understanding of this process is crucial for solving similar problems. The concept of exponents is used extensively in algebra and calculus, making it a vital skill to master.
To solve this, we multiply the fraction 3/4 by itself three times:
(3/4)^3 = (3/4) * (3/4) * (3/4)
When multiplying fractions, we multiply the numerators together and the denominators together:
Numerator: 3 * 3 * 3 = 27 Denominator: 4 * 4 * 4 = 64
Therefore, (3/4)^3 = 27/64.
This fraction, 27/64, is the final simplified form because 27 and 64 do not have any common factors other than 1. Understanding how to simplify fractions after raising them to a power is a vital step in ensuring accuracy and clarity in your solutions. The ability to simplify fractions not only provides the correct answer but also demonstrates a deeper understanding of mathematical principles.
In summary, to calculate (3/4)^3, we have systematically applied the definition of exponents by multiplying the fraction by itself the specified number of times and then simplifying the result. This method can be universally applied to any fraction raised to a positive integer power, making it a valuable technique in mathematical problem-solving. By breaking down the calculation into manageable steps, we can ensure accuracy and clarity in our solution, reinforcing the importance of methodical approaches in mathematics.
b. (-2/5)^5
Negative Fractions and Odd Powers: This problem introduces the concept of raising a negative fraction to a power. Specifically, we are tasked with calculating (-2/5)^5. When dealing with negative numbers raised to an odd power, the result will always be negative. This is because multiplying a negative number an odd number of times results in a negative product. Understanding this principle is crucial for accurately solving problems involving negative exponents and fractions.
To solve (-2/5)^5, we multiply the fraction -2/5 by itself five times:
(-2/5)^5 = (-2/5) * (-2/5) * (-2/5) * (-2/5) * (-2/5)
Since we are multiplying a negative fraction an odd number of times, the result will be negative. Let's calculate the numerical value:
Numerator: -2 * -2 * -2 * -2 * -2 = -32 Denominator: 5 * 5 * 5 * 5 * 5 = 3125
Therefore, (-2/5)^5 = -32/3125.
This fraction, -32/3125, is in its simplest form as 32 and 3125 do not share any common factors other than 1. The ability to recognize and work with negative fractions is a critical skill in algebra and calculus, as these types of problems frequently appear in more complex equations and applications. Accurate handling of negative signs is essential to avoid errors and achieve correct solutions.
In summary, the calculation of (-2/5)^5 highlights the importance of considering the sign when raising negative numbers to powers. By carefully applying the rules of exponents and multiplication for negative numbers, we can confidently solve such problems and ensure the accuracy of our results. This systematic approach not only provides the correct answer but also enhances our understanding of the fundamental principles governing mathematical operations.
c. (-5/8)^4
Negative Fractions and Even Powers: In this part, we calculate (-5/8)^4, which involves raising a negative fraction to an even power. When a negative number is raised to an even power, the result is always positive. This is because the negative signs cancel each other out in pairs during multiplication. Understanding this concept is crucial for correctly solving problems involving negative fractions and exponents.
To solve (-5/8)^4, we multiply the fraction -5/8 by itself four times:
(-5/8)^4 = (-5/8) * (-5/8) * (-5/8) * (-5/8)
Since we are multiplying a negative fraction an even number of times, the result will be positive. Let's calculate the numerical value:
Numerator: -5 * -5 * -5 * -5 = 625 Denominator: 8 * 8 * 8 * 8 = 4096
Therefore, (-5/8)^4 = 625/4096.
The fraction 625/4096 is already in its simplest form because 625 and 4096 do not share any common factors other than 1. The ability to recognize when a fraction is in its simplest form is an important skill in mathematical problem-solving, as it ensures that the final answer is presented in the most concise and accurate manner. This skill not only provides the correct answer but also demonstrates a deeper understanding of mathematical principles.
In summary, the calculation of (-5/8)^4 demonstrates the importance of understanding the effect of even powers on negative numbers. By applying the rules of exponents and multiplication correctly, we can confidently solve these types of problems and ensure the accuracy of our results. This systematic approach is vital for tackling more complex mathematical challenges and reinforces the significance of methodical problem-solving.
d. (4/5)^3
Positive Fractions and Powers: Now, we calculate (4/5)^3, which involves raising a positive fraction to a power. This operation is straightforward as there are no negative signs to consider. The focus here is on accurately multiplying the fraction by itself the specified number of times and simplifying the result. Understanding how to handle positive fractions raised to powers is a foundational skill in mathematics.
To solve (4/5)^3, we multiply the fraction 4/5 by itself three times:
(4/5)^3 = (4/5) * (4/5) * (4/5)
Multiply the numerators together and the denominators together:
Numerator: 4 * 4 * 4 = 64 Denominator: 5 * 5 * 5 = 125
Therefore, (4/5)^3 = 64/125.
The fraction 64/125 is in its simplest form as 64 and 125 do not share any common factors other than 1. The ability to recognize simplified fractions is crucial for presenting mathematical solutions in their most accurate and concise form. This skill not only ensures the correct answer but also demonstrates a solid grasp of fundamental mathematical principles.
In summary, the calculation of (4/5)^3 emphasizes the straightforward process of raising positive fractions to powers. By accurately applying the rules of exponents and multiplication, we can confidently solve such problems and ensure the precision of our results. This systematic approach is essential for building a strong foundation in mathematical operations.
e. (-7/10)^3
Negative Fractions and Odd Powers: In this section, we address (-7/10)^3, which involves raising a negative fraction to an odd power. As we previously discussed, the result will be negative when a negative number is raised to an odd power. This understanding is crucial for avoiding errors and ensuring the accuracy of our calculations. The principles of exponents and negative numbers are foundational in mathematics, making this a vital concept to master.
To solve (-7/10)^3, we multiply the fraction -7/10 by itself three times:
(-7/10)^3 = (-7/10) * (-7/10) * (-7/10)
Since we are multiplying a negative fraction an odd number of times, the result will be negative. Let's calculate the numerical value:
Numerator: -7 * -7 * -7 = -343 Denominator: 10 * 10 * 10 = 1000
Therefore, (-7/10)^3 = -343/1000.
The fraction -343/1000 is already in its simplest form because 343 and 1000 do not share any common factors other than 1. Presenting fractions in their simplest form is a key practice in mathematical solutions, as it demonstrates clarity and accuracy in understanding the problem. This skill not only provides the correct answer but also ensures that the solution is easily understood and interpreted.
In summary, the calculation of (-7/10)^3 reinforces the importance of considering the sign when raising negative numbers to odd powers. By meticulously applying the rules of exponents and multiplication for negative numbers, we can confidently solve these types of problems and achieve precise results. This systematic approach is fundamental for tackling more complex mathematical challenges.
f. (7/10)^3
Positive Fractions and Powers: Finally, we calculate (7/10)^3, which involves raising a positive fraction to a power. This is another straightforward calculation without the complexities of negative signs. The key here is to accurately multiply the fraction by itself the specified number of times and ensure the result is in its simplest form. Understanding how to handle positive fractions raised to powers is a basic yet essential skill in mathematics.
To solve (7/10)^3, we multiply the fraction 7/10 by itself three times:
(7/10)^3 = (7/10) * (7/10) * (7/10)
Multiply the numerators together and the denominators together:
Numerator: 7 * 7 * 7 = 343 Denominator: 10 * 10 * 10 = 1000
Therefore, (7/10)^3 = 343/1000.
The fraction 343/1000 is already in its simplest form because 343 and 1000 do not share any common factors other than 1. Recognizing simplified fractions is a critical aspect of presenting mathematical solutions accurately and concisely. This skill not only provides the correct answer but also showcases a thorough understanding of the problem and its solution.
In summary, the calculation of (7/10)^3 demonstrates the straightforward process of raising positive fractions to powers. By accurately applying the rules of exponents and multiplication, we can confidently solve such problems and ensure the precision of our results. This systematic approach is vital for building a solid foundation in mathematical operations.
This article has provided a step-by-step guide to calculating the values of rational numbers raised to powers. By understanding the principles of exponents and fractions, you can confidently solve these types of problems and enhance your mathematical skills. Mastering these concepts is crucial for success in more advanced mathematical studies.
