Question

For Exercises \(1-74,\) simplify. $$\frac{2^{4}+3(7+19)}{8^{2}-(2 \cdot 8+1)}$$

Short answer

Question: Simplify the expression $\frac{2^{4}+3(7+19)}{8^{2}-(2 \cdot 8+1)}$. Answer: The simplified expression is 2.

Step-by-step solution

Simplify terms inside parentheses

Simplify the expression inside the parentheses in the numerator: $$(7+19) = 26$$ Now the expression becomes: $$\frac{2^{4}+3(26)}{8^{2}-(2 \cdot 8+1)}$$

Simplify the exponentiation

Calculate the exponents: $$2^{4} = 16$$ $$8^{2} = 64$$ Now the expression becomes: $$\frac{16+3(26)}{64-(2 \cdot 8+1)}$$

Simplify terms inside parentheses

Simplify the expression inside the parentheses in the denominator: $$(2 \cdot 8+1) = 16+1 = 17$$ Now the expression becomes: $$\frac{16+3(26)}{64-17}$$

Perform multiplication

Multiply in the numerator: $$3(26) = 78$$ Now the expression becomes: $$\frac{16+78}{64-17}$$

Perform addition and subtraction

Perform addition in the numerator and subtraction in the denominator: $$16+78 = 94$$ $$64-17 = 47$$ Now the expression becomes: $$\frac{94}{47}$$

Simplify the fraction

Now, simplify the fraction by finding the greatest common divisor of both the numerator and the denominator. The greatest common divisor of 94 and 47 is 47. Divide both the numerator and the denominator by 47: $$\frac{94}{47} = \frac{94\div47}{47\div47} = \frac{2}{1}$$ So the simplified expression is: $$\frac{2}{1} = 2$$

Key concepts

Order of Operations

Understanding the order of operations, also known as the operator hierarchy, is crucial in simplifying mathematical expressions correctly. The order is generally remembered by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule tells us the sequence in which we should tackle operations within a complex expression to avoid mistakes.

For example, when presented with an expression like the one in our exercise \(\frac{2^{4}+3(7+19)}{8^{2}-(2 \cdot 8+1)}\), we must first address any calculations inside parentheses. This is why the terms \(7+19\) and \(2 \cdot 8+1\) were simplified first. Failure to follow the order can lead to different, incorrect results, a common pitfall for students working on algebraic expressions.

Exponentiation

Exponentiation is an operation involving two numbers, the base and the exponent. It indicates how many times the base is multiplied by itself. For instance, in the expression \(2^{4}\), 2 is the base and 4 is the exponent, meaning that 2 is multiplied by itself 4 times: \(2 \cdot 2 \cdot 2 \cdot 2 = 16\).

In our exercise, exponentiation is pivotal in determining the correct values for \(2^{4}\) and \(8^{2}\) before attempting any further arithmetic operations. Calculating these exponents correctly simplifies the expression significantly, allowing us to move on to other operations. Misunderstanding or incorrectly calculating exponents is another area that students often need clarification on, emphasizing the importance of mastering this operation.

Arithmetic Operations

Arithmetic operations, which include addition, subtraction, multiplication, and division, are the foundation of mathematics. Being comfortable with these operations is necessary for solving a wide array of problems. In expressions like the one in our exercise, we had to use all four arithmetic operations.

After handling parentheses and exponents, we moved on to multiplication, such as \(3(26)\), which simplifies to 78. This step is crucial as it collapses the expression, allowing us to perform simpler addition and subtraction next. We add figures within the numerator and subtract within the denominator to further simplify the expression. The final arithmetic operation in this problem is division to simplify the fraction \(\frac{94}{47}\), arriving at our final answer. These steps might seem straightforward, but each plays a significant role in the correct simplification of a complex expression. Always performing these with precision ensures accurate results.