Problem 9 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$\frac{16}{n}+2^{3}-10 \text { when } n=8$$

Step-by-Step Solution

Answer

The given expression evaluates to 0 when n=8.

Step 1: Substitute the given value of 'n' into the expression

Replace 'n' with '8' in the given expression. So, the expression becomes \(\frac{16}{8}+2^{3}-10\).

Step 2: Begin simplifying, respecting the BIDMAS/BODMAS rule

Simplify the fraction first since division comes before addition and subtraction in the order of operations. \(\frac{16}{8}=2\). So, the expression now reads as \(2+2^{3}-10\).

Step 3: Evaluate the power

Respecting the BIDMAS/BODMAS rule, calculate the exponent before addition/subtraction. \(2^{3} = 2*2*2 = 8\), hence the expression becomes \(2+8-10\).

Step 4: Simplify the expression

Now it only remains to carry out the addition and subtraction, from left to right. \(2+8=10\), so the expression transforms into \(10-10\). Subtracting 10 from 10, the final answer we get is 0.

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating algebraic expressions. This hierarchy, often remembered by the acronym PEMDAS or BODMAS, tells us the sequence to follow: Parentheses/Brackets first, Exponents/Orders (i.e., powers and roots, etc.) second, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). This order is key because performing operations in the wrong sequence can lead to incorrect results.

For example, in the expression \(\frac{16}{n}+2^{3}-10\) with \(n=8\), the order of operations tells us to calculate the division \(\frac{16}{8}\) and the exponent \(2^{3}\) before dealing with the addition and subtraction. This method ensures accuracy in finding that the expression simplifies correctly to yield a result of 0.

Substitution Method

The substitution method is a fundamental aspect of algebra which involves replacing variables with their given or known values. This is the first step in solving the algebraic expression.

Let's consider our example \(\frac{16}{n}+2^{3}-10\) where \(n=8\). We apply the substitution method by replacing \(n\) with 8, which simplifies the expression to \(\frac{16}{8}+2^{3}-10\). The substitution is straightforward, but it's essential to replace the variable consistently throughout the expression to avoid any computational mistakes.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form while maintaining their original value. This involves combining like terms, calculating variable values, and reducing fractions among other steps.

In the expression \(\frac{16}{8}+2^{3}-10\), simplification starts with resolving the fraction and the exponent. You first calculate the division \(\frac{16}{8}=2\), and then find the value of the exponent \(2^{3}=8\). After dealing with these two operations, the expression can be further simplified to \(2+8-10\), which is \(10-10\). Finally, subtracting 10 from 10 simplifies the entire expression to 0. The process of simplifying can include several other operations, depending on the complexity of the expression.

Exponents

Exponents represent the number of times a base is multiplied by itself. The expression \(2^{3}\) means \(2\) is multiplied three times: \(2*2*2\), which equals 8. Understanding how to evaluate exponents is vital when simplifying algebraic expressions that include them.

Importantly, when applying the order of operations, also known as BIDMAS/BODMAS, exponents are calculated after parentheses but before multiplication, division, addition, and subtraction. This concept is crucial in our example, as calculating the exponent \(2^{3}\) correctly is key to achieving the final simplified result of the expression.