Evaluate the expression inside the square root

Oct 13, 2025 by talynt.org 0 views

Simplifying Complex Expressions: A Step-by-Step Guide to Evaluating $7 \sqrt{3 \div \sqrt{10 + 3} }$

Mathematics is a vast and complex subject that involves solving various types of problems, including algebraic expressions. One of the most challenging aspects of mathematics is simplifying complex expressions, which often involve multiple operations and variables. In this article, we will focus on simplifying a specific complex expression: $7 \sqrt{3 \div \sqrt{10 + 3} }$ . We will break down the expression into smaller parts, apply the order of operations, and simplify it step by step.

The given expression is $7 \sqrt{3 \div \sqrt{10 + 3} }$ . To simplify this expression, we need to understand the order of operations, which is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is often remembered using the acronym PEMDAS, which stands for:

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

Let's break down the given expression into smaller parts and apply the order of operations:

Evaluate the expression inside the square root: The expression inside the square root is $10 + 3$ . To evaluate this expression, we need to add 10 and 3, which gives us $13$ .
Evaluate the square root: The square root of $13$ is a non-perfect square, so we cannot simplify it further.
Evaluate the division: The expression $3 \div \sqrt{13}$ involves a division operation. To evaluate this expression, we need to divide 3 by the square root of 13.
Evaluate the multiplication: The expression $7 \times 3 \div \sqrt{13}$ involves a multiplication operation. To evaluate this expression, we need to multiply 7 by 3 and then divide the result by the square root of 13.

Now that we have broken down the expression into smaller parts, let's simplify it step by step:

Evaluate the expression inside the square root: The expression inside the square root is $10 + 3$ , which equals $13$ .
Evaluate the square root: The square root of $13$ is approximately $3.61$ .
Evaluate the division: The expression $3 \div \sqrt{13}$ is approximately $3 \div 3.61$ , which equals $0.83$ .
Evaluate the multiplication: The expression $7 \times 3 \div \sqrt{13}$ is approximately $7 \times 0.83 \div 3.61$ , which equals $1.73$ .

In this article, we simplified a complex expression: $7 \sqrt{3 \div \sqrt{10 + 3} }$ . We broke down the expression into smaller parts, applied the order of operations, and simplified it step by step. The final simplified expression is approximately $1.73$ . We hope this article has provided you with a clear understanding of how to simplify complex expressions and has helped you develop your problem-solving skills.

When simplifying complex expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying complex expressions.
Not evaluating expressions inside parentheses first: Evaluate expressions inside parentheses first, as they take precedence over other operations.
Not simplifying fractions: Simplify fractions whenever possible to avoid unnecessary complexity.
Not checking for errors: Double-check your work for errors, especially when working with complex expressions.

Simplifying complex expressions has many real-world applications, including:

Science and engineering: Simplifying complex expressions is essential in science and engineering, where complex mathematical models are used to describe real-world phenomena.
Finance: Simplifying complex expressions is also important in finance, where complex financial models are used to predict stock prices and other financial metrics.
Computer programming: Simplifying complex expressions is a crucial skill for computer programmers, who often need to simplify complex mathematical expressions to write efficient code.

Simplifying complex expressions is a challenging but essential skill for anyone who wants to succeed in mathematics and other fields. By following the order of operations and breaking down complex expressions into smaller parts, we can simplify even the most complex expressions. We hope this article has provided you with a clear understanding of how to simplify complex expressions and has helped you develop your problem-solving skills.
Frequently Asked Questions: Simplifying Complex Expressions

Simplifying complex expressions is a crucial skill for anyone who wants to succeed in mathematics and other fields. In our previous article, we provided a step-by-step guide on how to simplify a complex expression: $7 \sqrt{3 \div \sqrt{10 + 3} }$ . In this article, we will answer some of the most frequently asked questions about simplifying complex expressions.

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is often remembered using the acronym PEMDAS, which stands for:

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

A: To simplify a complex expression, follow these steps:

Evaluate expressions inside parentheses first: Evaluate any expressions inside parentheses first, as they take precedence over other operations.
Evaluate any exponential expressions next: Evaluate any exponential expressions next, such as squaring or cubing.
Evaluate any multiplication and division operations from left to right: Evaluate any multiplication and division operations from left to right.
Finally, evaluate any addition and subtraction operations from left to right: Finally, evaluate any addition and subtraction operations from left to right.

A: A fraction is a way of expressing a part of a whole, such as 1/2 or 3/4. A decimal is a way of expressing a number as a sum of powers of 10, such as 0.5 or 0.75.

A: To simplify a fraction, follow these steps:

Find the greatest common divisor (GCD) of the numerator and denominator: Find the greatest common divisor (GCD) of the numerator and denominator.
Divide both the numerator and denominator by the GCD: Divide both the numerator and denominator by the GCD.
Simplify the fraction: Simplify the fraction by writing it in its simplest form.

A: A rational number is a number that can be expressed as a fraction, such as 1/2 or 3/4. An irrational number is a number that cannot be expressed as a fraction, such as the square root of 2 or the square root of 3.

A: To simplify a complex expression with multiple variables, follow these steps:

Identify the variables: Identify the variables in the expression.
Simplify the expression using the order of operations: Simplify the expression using the order of operations.
Combine like terms: Combine like terms to simplify the expression further.

A: Some common mistakes to avoid when simplifying complex expressions include:

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying complex expressions.
Not evaluating expressions inside parentheses first: Evaluate expressions inside parentheses first, as they take precedence over other operations.
Not simplifying fractions: Simplify fractions whenever possible to avoid unnecessary complexity.
Not checking for errors: Double-check your work for errors, especially when working with complex expressions.

Simplifying complex expressions is a crucial skill for anyone who wants to succeed in mathematics and other fields. By following the order of operations and breaking down complex expressions into smaller parts, we can simplify even the most complex expressions. We hope this article has provided you with a clear understanding of how to simplify complex expressions and has helped you develop your problem-solving skills.