Calculate The Total Of 4 1 3 + 3 2 5 − 2 14 / 15 4 \frac{1}{3} + 3 \frac{2}{5} - 2^{14/15} 4 3 1 ​ + 3 5 2 ​ − 2 14/15 .

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Introduction

In mathematics, we often encounter complex expressions that involve fractions, decimals, and exponents. Calculating these expressions can be challenging, but with a step-by-step approach, we can simplify them and arrive at the correct solution. In this article, we will focus on calculating the total of 413+325−214/154 \frac{1}{3} + 3 \frac{2}{5} - 2^{14/15}.

Understanding the Components

Before we dive into the calculation, let's break down the components of the expression:

  • 4134 \frac{1}{3}: This is a mixed number, which is a combination of a whole number and a fraction. In this case, the whole number is 4 and the fraction is 13\frac{1}{3}.
  • 3253 \frac{2}{5}: This is another mixed number, with a whole number of 3 and a fraction of 25\frac{2}{5}.
  • 214/152^{14/15}: This is an exponential expression, where 2 is the base and 1415\frac{14}{15} is the exponent.

Converting Mixed Numbers to Improper Fractions

To simplify the calculation, we need to convert the mixed numbers to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

  • 413=(4×3)+13=12+13=1334 \frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}
  • 325=(3×5)+25=15+25=1753 \frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}

Evaluating the Exponential Expression

Next, we need to evaluate the exponential expression 214/152^{14/15}. To do this, we can use the fact that 21/15=(21/15)142^{1/15} = (2^{1/15})^{14}. However, this is not a straightforward calculation, and we need to use a calculator or a computer program to evaluate it.

Using a Calculator or Computer Program

Using a calculator or computer program, we can evaluate 214/152^{14/15} as approximately 1.059.

Adding and Subtracting the Improper Fractions

Now that we have converted the mixed numbers to improper fractions and evaluated the exponential expression, we can add and subtract the fractions.

  • 133+175=65+5115=11615\frac{13}{3} + \frac{17}{5} = \frac{65 + 51}{15} = \frac{116}{15}
  • 11615−1.059=11615−15.88515=100.11515\frac{116}{15} - 1.059 = \frac{116}{15} - \frac{15.885}{15} = \frac{100.115}{15}

Simplifying the Result

Finally, we can simplify the result by dividing the numerator and denominator by their greatest common divisor (GCD).

  • 100.11515=6.6743...\frac{100.115}{15} = 6.6743...

Conclusion

In this article, we have calculated the total of 413+325−214/154 \frac{1}{3} + 3 \frac{2}{5} - 2^{14/15} using a step-by-step approach. We converted the mixed numbers to improper fractions, evaluated the exponential expression, added and subtracted the fractions, and simplified the result. The final answer is approximately 6.6743.

References

  • [1] "Mathematics for Dummies" by Mark Ryan
  • [2] "Algebra and Trigonometry" by Michael Sullivan
  • [3] "Calculus" by Michael Spivak

Glossary

  • Improper fraction: A fraction where the numerator is greater than or equal to the denominator.
  • Exponential expression: An expression that involves a base and an exponent.
  • Greatest common divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
    Calculating Complex Mathematical Expressions: A Q&A Guide ===========================================================

Introduction

In our previous article, we calculated the total of 413+325−214/154 \frac{1}{3} + 3 \frac{2}{5} - 2^{14/15} using a step-by-step approach. However, we know that math can be challenging, and sometimes we need a little extra help. In this article, we will answer some frequently asked questions about calculating complex mathematical expressions.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, 4134 \frac{1}{3} is a mixed number, while 133\frac{13}{3} is an improper fraction.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you need to multiply the whole number by the denominator and add the numerator. Then, divide the result by the denominator. For example, to convert 4134 \frac{1}{3} to an improper fraction, you would multiply 4 by 3 and add 1, then divide the result by 3: (4×3)+13=12+13=133\frac{(4 \times 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}.

Q: What is the order of operations for calculating complex mathematical expressions?

A: The order of operations is a set of rules that tells you which operations to perform first when calculating a mathematical expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, you need to raise the base to the power of the exponent. For example, to evaluate 214/152^{14/15}, you would raise 2 to the power of 1415\frac{14}{15}.

Q: What is the greatest common divisor (GCD) and how do I find it?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. To find the GCD, you can use the Euclidean algorithm or a calculator.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator and denominator by their greatest common divisor (GCD). For example, to simplify 100.11515\frac{100.115}{15}, you would divide the numerator and denominator by their GCD, which is 0.007: 100.11515=100.115÷0.00715÷0.007=14315.712142.86\frac{100.115}{15} = \frac{100.115 \div 0.007}{15 \div 0.007} = \frac{14315.71}{2142.86}.

Q: What are some common mistakes to avoid when calculating complex mathematical expressions?

A: Some common mistakes to avoid when calculating complex mathematical expressions include:

  • Not following the order of operations
  • Not converting mixed numbers to improper fractions
  • Not evaluating exponential expressions correctly
  • Not finding the greatest common divisor (GCD) when simplifying fractions

Conclusion

In this article, we have answered some frequently asked questions about calculating complex mathematical expressions. We hope that this guide has been helpful in clarifying any confusion and providing a better understanding of how to calculate complex mathematical expressions.

References

  • [1] "Mathematics for Dummies" by Mark Ryan
  • [2] "Algebra and Trigonometry" by Michael Sullivan
  • [3] "Calculus" by Michael Spivak

Glossary

  • Improper fraction: A fraction where the numerator is greater than or equal to the denominator.
  • Exponential expression: An expression that involves a base and an exponent.
  • Greatest common divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
  • Order of operations: A set of rules that tells you which operations to perform first when calculating a mathematical expression.