Problem 40 - Algebra 1

Question

Evaluate the expression. $$\frac{5^{3} \cdot 2}{1+6^{2}-8}$$

Step-by-Step Solution

Answer

The result of the expression \(\frac{5^{3} \cdot 2}{1+6^{2}-8}\) is approximately \(8.62\) when rounded to two decimal places.

Step 1: Solve Inside the Parentheses

The first step according to BIDMAS is to solve the operation inside parentheses. That gives us \(1+6^{2}-8\) which evaluates to \(1+36-8\).

Step 2: Solve the Resulting Expression

Perform the addition and subtraction on the set \(1+36-8\) to get \(29\).

Step 3: Evaluate the Numerator

The next step is to evaluate the numerator \(5^{3} \cdot 2\), which equals to \(125 \cdot 2 = 250\).

Step 4: Final Division

The final step is to solve the fraction by dividing the numerator by the denominator: \(\frac{250}{29}\)

Key Concepts

Order of Operations

Understanding the order of operations is critical when evaluating algebraic expressions. This rule is often remembered by the acronym PEMDAS or BIDMAS, which stands for Parentheses (or Brackets), Exponents (or Indices), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When faced with an expression such as \[\frac{5^{3} \times 2}{1+6^{2}-8}\], it's important to follow these steps in order. Start with any calculations inside parentheses or brackets, then move on to exponents, and so on. This ensures accurate results and is a foundational concept in algebra. By systematically working through the operations in the correct order, students can avoid common mistakes and build confidence in their problem-solving skills.

Exponentiation

Exponentiation is the process of raising a number to the power of an exponent. It represents repeated multiplication of the base number. In the expression \[5^{3}\], the base is 5 and the exponent is 3, meaning that 5 is multiplied by itself 3 times: \[5 \times 5 \times 5 = 125\].
Understanding how to handle exponents is crucial, as they often appear in algebraic expressions. Studying exponentiation includes learning about the laws of exponents, such as the product rule, power rule, and the rule for dividing exponents. This knowledge allows you to simplify complex expressions efficiently and is an integral part of mastering algebra.

Numerical Expression

A numerical expression is a mathematical phrase involving numbers and operation symbols, but no variables. In solving the numerical expression \[\frac{5^{3} \times 2}{1+6^{2}-8}\], you combine various arithmetic operations to evaluate the expression to a single number. After following the order of operations, you will first handle the exponentiation, then multiplication, and so on until you have simplified down to a single numerical value.

This process emphasizes the importance of understanding each element of the expression and their respective operations, which is an essential skill for tackling more complex algebraic expressions that include variables and unknowns. By mastering numerical expressions, you pave the way to solving more advanced mathematical problems.