[(3/5)⁰ + (3/5)¹ + (3/5)²] ÷ 7³/5³
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Introduction
In this article, we will delve into the world of mathematics and explore the concept of exponents and fractions. The given expression [(3/5)⁰ + (3/5)¹ + (3/5)²] ÷ 7³/5³ may seem complex at first glance, but with a step-by-step approach, we can break it down and understand its value.
Understanding Exponents
Exponents are a shorthand way of representing repeated multiplication. For example, 2⁴ can be read as "2 to the power of 4" or "2 multiplied by itself 4 times." In this case, we have exponents with fractions, such as (3/5)⁰, (3/5)¹, and (3/5)².
Zero Exponent
A zero exponent is a special case where the base is raised to the power of zero. In this case, (3/5)⁰ can be read as "3/5 to the power of 0." According to the rules of exponents, any number raised to the power of 0 is equal to 1. Therefore, (3/5)⁰ = 1.
Positive Exponents
Positive exponents represent repeated multiplication. For example, (3/5)¹ can be read as "3/5 multiplied by itself 1 time," which is equal to 3/5. Similarly, (3/5)² can be read as "3/5 multiplied by itself 2 times," which is equal to (3/5) × (3/5) = 9/25.
Simplifying the Expression
Now that we have understood the concept of exponents, let's simplify the given expression [(3/5)⁰ + (3/5)¹ + (3/5)²] ÷ 7³/5³.
Adding the Fractions
We can add the fractions (3/5)⁰, (3/5)¹, and (3/5)² by finding a common denominator. Since (3/5)⁰ = 1, we can rewrite the expression as (1 + 3/5 + 9/25) ÷ 7³/5³.
Finding a Common Denominator
To add the fractions, we need to find a common denominator. The least common multiple of 1, 5, and 25 is 25. Therefore, we can rewrite the expression as (25/25 + 15/25 + 9/25) ÷ 7³/5³.
Adding the Fractions
Now that we have a common denominator, we can add the fractions: (25/25 + 15/25 + 9/25) = 49/25.
Simplifying the Expression
Now that we have added the fractions, we can simplify the expression: (49/25) ÷ 7³/5³.
Dividing the Fractions
To divide the fractions, we need to invert the second fraction and multiply: (49/25) × (5³/7³).
Multiplying the Fractions
Now that we have inverted the second fraction, we can multiply the numerators and denominators: (49 × 5³) / (25 × 7³).
Evaluating the Expression
Now that we have multiplied the fractions, we can evaluate the expression: (49 × 125) / (25 × 343).
Simplifying the Expression
Now that we have evaluated the expression, we can simplify it: 6125 / 6825.
Reducing the Fraction
To reduce the fraction, we need to find the greatest common divisor of 6125 and 6825. The greatest common divisor is 25. Therefore, we can rewrite the fraction as (6125 ÷ 25) / (6825 ÷ 25).
Reducing the Fraction
Now that we have divided both the numerator and denominator by 25, we can simplify the fraction: 245 / 273.
Conclusion
In this article, we have explored the concept of exponents and fractions and simplified the given expression [(3/5)⁰ + (3/5)¹ + (3/5)²] ÷ 7³/5³. We have broken down the expression into smaller parts and used the rules of exponents and fractions to simplify it. The final answer is 245 / 273.
Final Answer
The final answer is 245 / 273.
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Introduction
In our previous article, we explored the concept of exponents and fractions and simplified the given expression [(3/5)⁰ + (3/5)¹ + (3/5)²] ÷ 7³/5³. In this article, we will answer some of the most frequently asked questions related to this topic.
Q&A
Q: What is the value of (3/5)⁰?
A: According to the rules of exponents, any number raised to the power of 0 is equal to 1. Therefore, (3/5)⁰ = 1.
Q: What is the value of (3/5)¹?
A: (3/5)¹ can be read as "3/5 multiplied by itself 1 time," which is equal to 3/5.
Q: What is the value of (3/5)²?
A: (3/5)² can be read as "3/5 multiplied by itself 2 times," which is equal to (3/5) × (3/5) = 9/25.
Q: How do you add fractions with different denominators?
A: To add fractions with different denominators, you need to find a common denominator. The least common multiple of the denominators is the common denominator.
Q: How do you simplify a fraction?
A: To simplify a fraction, you need to find the greatest common divisor of the numerator and denominator. Divide both the numerator and denominator by the greatest common divisor to simplify the fraction.
Q: What is the value of (49/25) ÷ 7³/5³?
A: To divide the fractions, you need to invert the second fraction and multiply: (49/25) × (5³/7³). Multiply the numerators and denominators: (49 × 5³) / (25 × 7³). Evaluate the expression: (49 × 125) / (25 × 343). Simplify the expression: 6125 / 6825. Reduce the fraction: 245 / 273.
Q: What is the final answer to the expression [(3/5)⁰ + (3/5)¹ + (3/5)²] ÷ 7³/5³?
A: The final answer is 245 / 273.
Common Mistakes
Mistake 1: Not understanding the concept of exponents
Exponents are a shorthand way of representing repeated multiplication. Make sure you understand the concept of exponents before attempting to simplify the expression.
Mistake 2: Not finding a common denominator
To add fractions with different denominators, you need to find a common denominator. The least common multiple of the denominators is the common denominator.
Mistake 3: Not simplifying the fraction
To simplify a fraction, you need to find the greatest common divisor of the numerator and denominator. Divide both the numerator and denominator by the greatest common divisor to simplify the fraction.
Conclusion
In this article, we have answered some of the most frequently asked questions related to the expression [(3/5)⁰ + (3/5)¹ + (3/5)²] ÷ 7³/5³. We have also discussed some common mistakes that people make when simplifying this expression. Make sure you understand the concept of exponents and fractions before attempting to simplify this expression.
Final Answer
The final answer is 245 / 273.