Problem 75 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ \left(\frac{6+5 \cdot 0}{6}\right)^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Simplify the expression inside the parentheses

First, evaluate the expression inside the parentheses. According to the order of operations (PEMDAS/BODMAS), calculate the multiplication first. Here, we compute the multiplication \(5 \times 0\) which equals \(0\). So the expression becomes: \(\frac{6 + 0}{6}\)

Step 2: Add the numbers in the numerator

Add the remaining numbers in the numerator: \(6 + 0 = 6\). Therefore, the expression simplifies to: \(\frac{6}{6}\)

Step 3: Divide the numerator by the denominator

Divide the numerator by the denominator: \(\frac{6}{6} = 1\). So the expression inside the parentheses simplifies to \(1\).

Step 4: Square the result

Finally, square the result: \(1^2 = 1\).

Key Concepts

Order of Operations

When evaluating any mathematical expression, it's essential to follow the correct order of operations. This ensures you get the right answer every time. The standard order is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), often remembered by the acronym PEMDAS (or BODMAS in some regions).

In our exercise, we start by simplifying the expression inside the parentheses: \(\frac{6+5 \times 0}{6}\). Following PEMDAS, multiplication comes before addition. Therefore, we compute 5 * 0 first and then add 6.

Simplifying Fractions

Simplifying fractions is crucial for making expressions easier to handle. A simplified fraction has the numerator and denominator reduced to their smallest values that still represent the same ratio.

Within our problem, once we have \(\frac{6 + 0}{6}\), we compute the addition in the numerator: 6 + 0 = 6. This gives us \(\frac{6}{6}\). A fraction where the numerator and the denominator are equal simplifies to 1, as a number divided by itself equals 1. Thus, \(\frac{6}{6} = 1\).

Basic Algebra

Understanding basic algebra is crucial for breaking down and solving expressions. Algebra helps us manipulate mathematical symbols and structures to find unknown values or simplify expressions.

In this exercise, knowledge of basic algebraic rules helps us to know to reduce the fraction \(\frac{6+0}{6}\) by performing the operations inside the parentheses (division here) to make the fraction \(\frac{6}{6}\) simplify to 1. Mastering these basic principles allows us to solve more complex algebraic problems with confidence.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. The notation \(a^n\) means 'a' is multiplied by itself 'n' times.

In the given problem, after simplifying the expression inside the parentheses to 1, we need to handle the exponent: \(1^2\). Raising any number to the power of 2 means multiplying that number by itself. So, \(1^2 = 1 \times 1\), which equals 1.

Recognizing how to handle exponents correctly is important for evaluating expressions accurately and efficiently.