Evaluate The Expression:$\[ \left(\frac{7}{2} - \frac{4}{3} + \frac{1}{8}\right) \cdot \sqrt{\frac{1}{2} \div \frac{2}{9}} + 3^{-2} \\]

Introduction

In mathematics, evaluating expressions is a fundamental concept that involves simplifying complex mathematical expressions to obtain a final value. These expressions can be algebraic, trigonometric, or a combination of both, and they often involve various mathematical operations such as addition, subtraction, multiplication, and division. In this article, we will evaluate the expression $\left(\frac{7}{2} - \frac{4}{3} + \frac{1}{8}\right) \cdot \sqrt{\frac{1}{2} \div \frac{2}{9}} + 3^{-2}$ and provide a step-by-step guide on how to simplify it.

Understanding the Expression

The given expression involves several mathematical operations, including addition, subtraction, multiplication, and division. It also involves the use of square roots and exponents. To simplify this expression, we need to follow the order of operations (PEMDAS), which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Step 1: Evaluate the Expression Inside the Parentheses

The first step is to evaluate the expression inside the parentheses, which is $\frac{7}{2} - \frac{4}{3} + \frac{1}{8}$. To do this, we need to find a common denominator for the fractions. The least common multiple (LCM) of 2, 3, and 8 is 24.

\frac{7}{2} = \frac{7 \cdot 12}{2 \cdot 12} = \frac{84}{24}
\\]
\frac{4}{3} = \frac{4 \cdot 8}{3 \cdot 8} = \frac{32}{24}
\\]
\frac{1}{8} = \frac{1 \cdot 3}{8 \cdot 3} = \frac{3}{24}
\\]

Step 2: Subtract and Add the Fractions

Now that we have a common denominator, we can subtract and add the fractions.

\frac{84}{24} - \frac{32}{24} + \frac{3}{24} = \frac{55}{24}
\\]

Step 3: Evaluate the Square Root

The next step is to evaluate the square root of $\frac{1}{2} \div \frac{2}{9}$. To do this, we need to follow the order of operations and first evaluate the division.

\frac{1}{2} \div \frac{2}{9} = \frac{1}{2} \cdot \frac{9}{2} = \frac{9}{4}
\\]

Step 4: Evaluate the Square Root of $\frac{9}{4}$

Now that we have the value of $\frac{9}{4}$, we can evaluate the square root.

\sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}
\\]

Step 5: Multiply the Two Values

The next step is to multiply the two values, $\frac{55}{24}$ and $\frac{3}{2}$.

\frac{55}{24} \cdot \frac{3}{2} = \frac{55 \cdot 3}{24 \cdot 2} = \frac{165}{48}
\\]

Step 6: Simplify the Fraction

The final step is to simplify the fraction $\frac{165}{48}$.

\frac{165}{48} = \frac{165 \div 3}{48 \div 3} = \frac{55}{16}
\\]

Step 7: Evaluate the Exponent

The final step is to evaluate the exponent $3^{-2}$.

3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\\]

Step 8: Add the Two Values

The final step is to add the two values, $\frac{55}{16}$ and $\frac{1}{9}$.

\frac{55}{16} + \frac{1}{9} = \frac{55 \cdot 9}{16 \cdot 9} + \frac{1 \cdot 16}{9 \cdot 16} = \frac{495}{144} + \frac{16}{144} = \frac{511}{144}
\\]

Conclusion

In conclusion, the final value of the expression $\left(\frac{7}{2} - \frac{4}{3} + \frac{1}{8}\right) \cdot \sqrt{\frac{1}{2} \div \frac{2}{9}} + 3^{-2}$ is $\frac{511}{144}$. This expression involves several mathematical operations, including addition, subtraction, multiplication, and division, as well as the use of square roots and exponents. By following the order of operations and simplifying the expression step-by-step, we were able to obtain the final value.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Q: How do I simplify a fraction? A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.
• Q: How do I evaluate an exponent? A: To evaluate an exponent, you need to raise the base number to the power of the exponent.

Final Thoughts

Evaluating expressions is a fundamental concept in mathematics that involves simplifying complex mathematical expressions to obtain a final value. By following the order of operations and simplifying the expression step-by-step, we can obtain the final value of the expression. In this article, we evaluated the expression $\left(\frac{7}{2} - \frac{4}{3} + \frac{1}{8}\right) \cdot \sqrt{\frac{1}{2} \div \frac{2}{9}} + 3^{-2}$ and obtained the final value of $\frac{511}{144}$.

Introduction

In our previous article, we evaluated the expression $\left(\frac{7}{2} - \frac{4}{3} + \frac{1}{8}\right) \cdot \sqrt{\frac{1}{2} \div \frac{2}{9}} + 3^{-2}$ and obtained the final value of $\frac{511}{144}$. In this article, we will answer some frequently asked questions about evaluating expressions and simplifying complex mathematical expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This means that you need to follow this order when evaluating an expression:

1. Evaluate expressions inside parentheses
2. Evaluate exponents
3. Evaluate multiplication and division from left to right
4. Evaluate addition and subtraction from left to right

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD. For example, to simplify the fraction $\frac{12}{16}$, you need to find the GCD of 12 and 16, which is 4. Then, you can divide both numbers by 4 to get $\frac{3}{4}$.

Q: How do I evaluate an exponent?

A: To evaluate an exponent, you need to raise the base number to the power of the exponent. For example, to evaluate the expression $2^3$, you need to raise 2 to the power of 3, which is equal to 8.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change. For example, in the expression $x + 5$, x is a variable because its value can change, while 5 is a constant because its value does not change.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to multiply the numerator and denominator by the reciprocal of the denominator. For example, to simplify the fraction $\frac{\frac{1}{2}}{\frac{3}{4}}$, you need to multiply the numerator and denominator by the reciprocal of the denominator, which is $\frac{4}{3}$. This gives you $\frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3}$.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction, while an irrational expression is an expression that cannot be written as a fraction. For example, the expression $\frac{1}{2}$ is a rational expression because it can be written as a fraction, while the expression $\sqrt{2}$ is an irrational expression because it cannot be written as a fraction.

Q: How do I evaluate a square root?

A: To evaluate a square root, you need to find the number that, when multiplied by itself, gives the original number. For example, to evaluate the expression $\sqrt{16}$, you need to find the number that, when multiplied by itself, gives 16, which is 4.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression that can be written in the form $ax + b$, where a and b are constants, while a quadratic expression is an expression that can be written in the form $ax^2 + bx + c$, where a, b, and c are constants. For example, the expression $2x + 3$ is a linear expression because it can be written in the form $ax + b$, while the expression $x^2 + 2x + 1$ is a quadratic expression because it can be written in the form $ax^2 + bx + c$.

Conclusion

In conclusion, evaluating expressions and simplifying complex mathematical expressions is an important concept in mathematics that involves following the order of operations and simplifying the expression step-by-step. By following the order of operations and simplifying the expression, we can obtain the final value of the expression. In this article, we answered some frequently asked questions about evaluating expressions and simplifying complex mathematical expressions.

Final Thoughts

Evaluating expressions and simplifying complex mathematical expressions is a fundamental concept in mathematics that involves following the order of operations and simplifying the expression step-by-step. By following the order of operations and simplifying the expression, we can obtain the final value of the expression. In this article, we answered some frequently asked questions about evaluating expressions and simplifying complex mathematical expressions. We hope that this article has been helpful in understanding the concept of evaluating expressions and simplifying complex mathematical expressions.

Additional Resources

• Mathway: A online math problem solver that can help you solve math problems and evaluate expressions.
• Khan Academy: A online learning platform that provides video lessons and practice exercises on various math topics, including evaluating expressions and simplifying complex mathematical expressions.
• Math Open Reference: A online math reference book that provides detailed explanations and examples of various math topics, including evaluating expressions and simplifying complex mathematical expressions.