Calculate The Expression:$\[ \sqrt[3]{\frac{1}{64} \div \frac{27}{8}} \cdot \sqrt{\left(\frac{1}{5}\right)^2} + 5 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 \\]

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Introduction

In this article, we will delve into the world of mathematics and solve a complex expression involving exponents, roots, and fractions. The expression is given as:

{ \sqrt[3]{\frac{1}{64} \div \frac{27}{8}} \cdot \sqrt{\left(\frac{1}{5}\right)^2} + 5 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 \}$ \] We will break down the expression into smaller parts, solve each part, and then combine the results to obtain the final answer. **Step 1: Simplify the Fraction** ------------------------------- The first step is to simplify the fraction $\frac{1}{64} \div \frac{27}{8}$. To do this, we can use the rule for dividing fractions, which states that $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$. ```python from fractions import Fraction # Define the fractions frac1 = Fraction(1, 64) frac2 = Fraction(27, 8) # Divide the fractions result = frac1 / frac2 print(result) ``` This will output the result of the division, which is $\frac{1}{64} \div \frac{27}{8} = \frac{1}{64} \cdot \frac{8}{27} = \frac{1}{216}$. **Step 2: Simplify the Root** --------------------------- Next, we need to simplify the cube root of $\frac{1}{216}$. To do this, we can use the rule for cube roots, which states that $\sqrt[3]{a} = a^{\frac{1}{3}}$. ```python import math # Define the number num = Fraction(1, 216) # Calculate the cube root result = num ** (1/3) print(result) ``` This will output the result of the cube root, which is $\sqrt[3]{\frac{1}{216}} = \frac{1}{6}$. **Step 3: Simplify the Square Root** --------------------------------- Now, we need to simplify the square root of $\left(\frac{1}{5}\right)^2$. To do this, we can use the rule for square roots, which states that $\sqrt{a^2} = a$. ```python import math # Define the number num = Fraction(1, 5) ** 2 # Calculate the square root result = num ** (1/2) print(result) ``` This will output the result of the square root, which is $\sqrt{\left(\frac{1}{5}\right)^2} = \frac{1}{5}$. **Step 4: Simplify the Exponents** ------------------------------- Next, we need to simplify the exponents $\left(\frac{1}{2}\right)^2$ and $\left(\frac{1}{2}\right)^3$. To do this, we can use the rule for exponents, which states that $a^m \cdot a^n = a^{m+n}$. ```python import math # Define the numbers num1 = Fraction(1, 2) ** 2 num2 = Fraction(1, 2) ** 3 # Calculate the exponents result1 = num1 result2 = num2 print(result1) print(result2) ``` This will output the results of the exponents, which are $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ and $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$. **Step 5: Combine the Results** --------------------------- Finally, we can combine the results of the previous steps to obtain the final answer. ```python import math # Define the numbers num1 = Fraction(1, 6) num2 = Fraction(1, 5) num3 = Fraction(1, 4) num4 = Fraction(1, 8) # Calculate the final answer result = num1 * num2 + 5 + num3 + num4 print(result) ``` This will output the final answer, which is $\frac{1}{30} + 5 + \frac{1}{4} + \frac{1}{8} = \frac{1}{30} + \frac{5}{1} + \frac{2}{8} + \frac{1}{8} = \frac{1}{30} + \frac{200}{30} + \frac{3}{30} + \frac{1}{30} = \frac{205}{30} = 6.83333333333$. **Conclusion** ---------- In this article, we have solved a complex expression involving exponents, roots, and fractions. We have broken down the expression into smaller parts, solved each part, and then combined the results to obtain the final answer. The final answer is $\frac{205}{30} = 6.83333333333$.<br/> **Frequently Asked Questions: Solving the Expression** ===================================================== **Q: What is the main concept behind solving the expression?** -------------------------------------------------------- A: The main concept behind solving the expression is to break it down into smaller parts, solve each part, and then combine the results to obtain the final answer. This involves using various mathematical rules and formulas, such as the rule for dividing fractions, the rule for cube roots, and the rule for exponents. **Q: What is the first step in solving the expression?** ------------------------------------------------ A: The first step in solving the expression is to simplify the fraction $\frac{1}{64} \div \frac{27}{8}$. This involves using the rule for dividing fractions, which states that $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$. **Q: How do I simplify the cube root of $\frac{1}{216}$?** -------------------------------------------------------- A: To simplify the cube root of $\frac{1}{216}$, you can use the rule for cube roots, which states that $\sqrt[3]{a} = a^{\frac{1}{3}}$. This involves raising the number to the power of $\frac{1}{3}$. **Q: How do I simplify the square root of $\left(\frac{1}{5}\right)^2$?** ------------------------------------------------------------------- A: To simplify the square root of $\left(\frac{1}{5}\right)^2$, you can use the rule for square roots, which states that $\sqrt{a^2} = a$. This involves taking the square root of the number. **Q: How do I simplify the exponents $\left(\frac{1}{2}\right)^2$ and $\left(\frac{1}{2}\right)^3$?** ----------------------------------------------------------------------------------------------- A: To simplify the exponents $\left(\frac{1}{2}\right)^2$ and $\left(\frac{1}{2}\right)^3$, you can use the rule for exponents, which states that $a^m \cdot a^n = a^{m+n}$. This involves raising the number to the power of the exponent. **Q: How do I combine the results of the previous steps to obtain the final answer?** -------------------------------------------------------------------------------- A: To combine the results of the previous steps, you can use the rules for addition and multiplication. This involves adding and multiplying the numbers together to obtain the final answer. **Q: What is the final answer to the expression?** ------------------------------------------------ A: The final answer to the expression is $\frac{205}{30} = 6.83333333333$. **Q: What are some common mistakes to avoid when solving the expression?** --------------------------------------------------------------------- A: Some common mistakes to avoid when solving the expression include: * Not simplifying the fractions and exponents correctly * Not using the correct rules for dividing fractions, cube roots, and exponents * Not combining the results of the previous steps correctly * Not checking the final answer for accuracy **Q: How can I practice solving expressions like this one?** --------------------------------------------------------- A: You can practice solving expressions like this one by: * Working through example problems and exercises * Using online resources and calculators to check your work * Asking a teacher or tutor for help and guidance * Joining a study group or online community to discuss and learn from others **Q: What are some real-world applications of solving expressions like this one?** ------------------------------------------------------------------------- A: Some real-world applications of solving expressions like this one include: * Calculating the area and perimeter of shapes and figures * Determining the cost and profit of a business or investment * Solving problems in physics, engineering, and other fields that involve mathematical calculations * Creating and solving mathematical models to predict and analyze real-world phenomena.