Evaluate:$\[ \frac{12 + 7 \cdot 6}{(8 - 5)^2} \\]

Mar 1, 2025 by ADMIN 50 views

$Evaluate: $\frac{12 + 7 \cdot 6}{(8 - 5)^2}$$

Introduction

In mathematics, evaluating expressions is a crucial skill that involves simplifying complex mathematical expressions to obtain a final value. This skill is essential in various mathematical operations, including algebra, geometry, and calculus. In this article, we will evaluate the given expression $\frac{12 + 7 \cdot 6}{(8 - 5)^2}$ and provide a step-by-step solution to simplify it.

Understanding the Expression

The given expression is a fraction, which consists of two parts: the numerator and the denominator. The numerator is $12 + 7 \cdot 6$ , and the denominator is $(8 - 5)^2$ . To evaluate this expression, we need to follow the order of operations (PEMDAS):

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

Evaluating the Numerator

The numerator is $12 + 7 \cdot 6$ . To evaluate this expression, we need to follow the order of operations:

Multiply 7 and 6: $7 \cdot 6 = 42$
Add 12 and 42: $12 + 42 = 54$

So, the numerator is equal to 54.

Evaluating the Denominator

The denominator is $(8 - 5)^2$ . To evaluate this expression, we need to follow the order of operations:

Subtract 5 from 8: $8 - 5 = 3$
Square 3: $3^2 = 9$

So, the denominator is equal to 9.

Evaluating the Expression

Now that we have evaluated the numerator and the denominator, we can substitute these values into the original expression:

$\frac{12 + 7 \cdot 6}{(8 - 5)^2} = \frac{54}{9}$

Simplifying the Expression

To simplify the expression, we can divide the numerator by the denominator:

$\frac{54}{9} = 6$

So, the final value of the expression is 6.

Conclusion

In this article, we evaluated the given expression $\frac{12 + 7 \cdot 6}{(8 - 5)^2}$ and provided a step-by-step solution to simplify it. We followed the order of operations (PEMDAS) to evaluate the numerator and the denominator, and finally, we simplified the expression to obtain a final value of 6. This example demonstrates the importance of following the order of operations and simplifying complex mathematical expressions to obtain a final value.

Frequently Asked Questions

What is the order of operations (PEMDAS)? The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
How do I simplify complex mathematical expressions? To simplify complex mathematical expressions, you need to follow the order of operations (PEMDAS) and evaluate the expression step-by-step. This may involve multiplying, dividing, adding, or subtracting numbers, as well as evaluating expressions inside parentheses.

Additional Resources

Khan Academy: Order of Operations
Mathway: Simplifying Expressions
Wolfram Alpha: Evaluating Expressions

Final Thoughts

Evaluating expressions is a crucial skill in mathematics that involves simplifying complex mathematical expressions to obtain a final value. By following the order of operations (PEMDAS) and simplifying expressions step-by-step, you can obtain accurate results and solve mathematical problems with confidence.

Introduction

Evaluating expressions is a fundamental concept in mathematics that involves simplifying complex mathematical expressions to obtain a final value. In our previous article, we evaluated the expression $\frac{12 + 7 \cdot 6}{(8 - 5)^2}$ and provided a step-by-step solution to simplify it. In this article, we will address some of the most frequently asked questions about evaluating expressions and provide additional resources for further learning.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for: + Parentheses: Evaluate expressions inside parentheses first. + Exponents: Evaluate any exponential expressions next. + Multiplication and Division: Evaluate multiplication and division operations from left to right. + Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify complex mathematical expressions?

A: To simplify complex mathematical expressions, you need to follow the order of operations (PEMDAS) and evaluate the expression step-by-step. This may involve multiplying, dividing, adding, or subtracting numbers, as well as evaluating expressions inside parentheses.

Q: What is the difference between an expression and an equation?

A: An expression is a mathematical statement that contains variables, constants, and mathematical operations. An equation is a mathematical statement that contains an expression on each side of the equals sign. For example, $x + 3 = 5$ is an equation, while $x + 3$ is an expression.

Q: How do I evaluate expressions with variables?

A: To evaluate expressions with variables, you need to substitute a value for the variable and then simplify the expression. For example, if you have the expression $2x + 3$ and you substitute $x = 4$ , you would evaluate the expression as follows: + Substitute $x = 4$ into the expression: $2(4) + 3$ + Simplify the expression: $8 + 3 = 11$

Q: What is the difference between a numerical expression and an algebraic expression?

A: A numerical expression is a mathematical statement that contains only numbers and mathematical operations. An algebraic expression is a mathematical statement that contains variables, constants, and mathematical operations. For example, $2 + 3$ is a numerical expression, while $2x + 3$ is an algebraic expression.

Q: How do I evaluate expressions with fractions?

A: To evaluate expressions with fractions, you need to follow the order of operations (PEMDAS) and simplify the expression. This may involve multiplying, dividing, adding, or subtracting fractions, as well as evaluating expressions inside parentheses.

Additional Resources

Khan Academy: Order of Operations
Mathway: Simplifying Expressions
Wolfram Alpha: Evaluating Expressions
MIT OpenCourseWare: Algebra
Math Is Fun: Evaluating Expressions