Evaluate The Following Expression:${ 8 \times \left(\frac{3^2}{12} + 5 + \left(13 + 2^4\right)\right) }$

Introduction

In this article, we will evaluate the given mathematical expression, which involves various operations such as exponentiation, division, addition, and multiplication. The expression is: ${ 8 \times \left(\frac{3^2}{12} + 5 + \left(13 + 2^4\right)\right) }$. We will break down the expression step by step, following the order of operations, to simplify it and find the final result.

Step 1: Evaluate the Exponents

The first step is to evaluate the exponents in the expression. We have two exponents: $3^2$ and $2^4$. Let's calculate them:

• $3^2 = 3 \times 3 = 9$
• $2^4 = 2 \times 2 \times 2 \times 2 = 16$

Step 2: Simplify the Fractions

Next, we need to simplify the fraction $\frac{3^2}{12}$. We can do this by dividing the numerator by the denominator:

• $\frac{3^2}{12} = \frac{9}{12} = \frac{3}{4}$

Step 3: Evaluate the Expression Inside the Parentheses

Now, let's evaluate the expression inside the parentheses: $\left(13 + 2^4\right)$. We already calculated $2^4 = 16$ in Step 1. So, we can substitute this value into the expression:

• $\left(13 + 2^4\right) = \left(13 + 16\right) = 29$

Step 4: Add the Terms Inside the Parentheses

Next, we need to add the terms inside the parentheses: $\frac{3}{4} + 5 + 29$. Let's calculate this:

• $\frac{3}{4} + 5 + 29 = \frac{3}{4} + 34 = 34 + \frac{3}{4} = 34 + 0.75 = 34.75$

Step 5: Multiply the Result by 8

Finally, we need to multiply the result by 8:

• $8 \times 34.75 = 278$

The final answer is $\boxed{278}$.

Conclusion

In this article, we evaluated the given mathematical expression step by step, following the order of operations. We simplified the fractions, evaluated the exponents, and added the terms inside the parentheses. Finally, we multiplied the result by 8 to get the final answer. The expression $8 \times \left(\frac{3^2}{12} + 5 + \left(13 + 2^4\right)\right)$ simplifies to $\boxed{278}$.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Q: How do I simplify fractions? A: To simplify a fraction, we need to divide the numerator by the denominator. If the result is a whole number, we can write it as a whole number. If the result is a decimal, we can write it as a decimal.
• Q: How do I evaluate expressions with exponents? A: To evaluate an expression with an exponent, we need to multiply the base by itself as many times as the exponent indicates. For example, $2^3 = 2 \times 2 \times 2 = 8$.

Related Topics

• Exponents and Powers
• Fractions and Decimals
• Order of Operations
• Algebraic Expressions

Further Reading

• Khan Academy: Order of Operations
• Mathway: Exponents and Powers
• Purplemath: Fractions and Decimals
  

Introduction

In our previous article, we evaluated the mathematical expression $8 \times \left(\frac{3^2}{12} + 5 + \left(13 + 2^4\right)\right)$. In this article, we will answer some frequently asked questions related to evaluating mathematical expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
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 simplify fractions?

A: To simplify a fraction, we need to divide the numerator by the denominator. If the result is a whole number, we can write it as a whole number. If the result is a decimal, we can write it as a decimal.

For example, to simplify the fraction $\frac{12}{4}$, we can divide the numerator by the denominator:

$\frac{12}{4} = 3$

So, the simplified fraction is $3$.

Q: How do I evaluate expressions with exponents?

A: To evaluate an expression with an exponent, we need to multiply the base by itself as many times as the exponent indicates. For example, $2^3 = 2 \times 2 \times 2 = 8$.

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. For example, $x$ is a variable.

A constant, on the other hand, is a value that does not change. For example, $5$ is a constant.

Q: How do I evaluate expressions with multiple operations?

A: To evaluate an expression with multiple operations, we need to follow the order of operations. For example, to evaluate the expression $3 + 2 \times 4$, we need to follow the order of operations:

1. Multiply $2$ and $4$: $2 \times 4 = 8$
2. Add $3$ and $8$: $3 + 8 = 11$

So, the final result is $11$.

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

A: An equation is a statement that says two expressions are equal. For example, $x + 2 = 5$ is an equation.

An expression, on the other hand, is a group of numbers, variables, and operations that can be evaluated to a single value. For example, $x + 2$ is an expression.

Conclusion

In this article, we answered some frequently asked questions related to evaluating mathematical expressions. We covered topics such as the order of operations, simplifying fractions, evaluating expressions with exponents, and the difference between variables and constants. We also discussed how to evaluate expressions with multiple operations and the difference between equations and expressions.

Frequently Asked Questions

• Q: What is the order of operations? A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression.
• Q: How do I simplify fractions? A: To simplify a fraction, we need to divide the numerator by the denominator.
• Q: How do I evaluate expressions with exponents? A: To evaluate an expression with an exponent, we need to multiply the base by itself as many times as the exponent indicates.
• 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.

Related Topics

• Exponents and Powers
• Fractions and Decimals
• Order of Operations
• Algebraic Expressions

Further Reading

• Khan Academy: Order of Operations
• Mathway: Exponents and Powers
• Purplemath: Fractions and Decimals