Problem 38 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Then simplify the answer. $$ \frac{5^{2} \cdot 2}{1+6^{2}-12} $$

Step-by-Step Solution

Answer

The simplified answer of the expression is 2.

Step 1: Perform exponentiation

Calculate \(5^{2}\) and \(6^{2}\), which gives 25 and 36 respectively. So, the expression becomes \(\frac{25 \cdot 2}{1+36-12}\).

Step 2: Perform multiplication

Next, calculate the multiplication which gives 50. So, the expression becomes \(\frac{50}{1+36-12}\).

Step 3: Perform addition and subtraction

Now, calculate the addition and subtraction in the denominator to get 25. So, the expression becomes \(\frac{50}{25}\).

Step 4: Simplify the fraction

Perform the division operation to get the simplified answer. So, \(\frac{50}{25} = 2\).

Key Concepts

Exponentiation

Understanding exponentiation is critical when working with algebraic expressions. In the context of simplifying expressions, it involves raising numbers to a power, which means multiplying the number by itself a certain number of times. For instance, in the expression \(5^{2}\), the base 5 is multiplied by itself to give 25. Similarly, \(6^{2}\) means 6 multiplied by 6, yielding 36. This step is crucial because it affects the values you’ll work with in the subsequent steps of simplification.

• Exponentiation comes before any other operation except parentheses in the order of operations.
• Understand the 'power of a power' rule and 'negative exponents' for more complex expressions.
• Remember that \(a^{0}=1\) for any non-zero \(a\), and \(a^{1}=a\).

It is important to evaluate these powers correctly to proceed with the simplification process accurately.

Numerical Expression

A numerical expression is a mathematical phrase that can contain numbers and operation symbols but no variables. To evaluate such expressions, like \(\frac{5^{2} \cdot 2}{1+6^{2}-12}\), you need to understand how to combine these numbers using the given operations. The goal is to turn an expression with many components into a single numerical value. When encountering a complex numerical expression:

• Identify and carry out operations in 'terms', which are parts of the expression separated by plus or minus signs.
• Look out for parentheses or brackets, as they indicate which operations should be performed first.
• Once exponents are evaluated, and multiplication or division is performed, add or subtract ‘like terms’ if necessary to find your final solution.

Evaluating a numerical expression step by step can help to avoid mistakes and ensure the accuracy of your final answer.

Order of Operations

• PEMDAS/BODMAS is a common acronym/mnemonic to remember the order: Parentheses/Brackets, Exponents/Orders, Multiplication/Division (from left to right), and Addition/Subtraction (from left to right).
• Always work through operations inside parentheses or brackets first.
• Look out for special cases, such as negative numbers and zero.

In our example, after calculating exponents, multiplication was performed before tackling the addition and subtraction in the denominator. Once these were simplified, the final fraction was divided to yield the answer 2. This structured approach ensures a clear path to the correct simplified form of an algebraic expression. Embracing the order of operations is essential because it guarantees that all mathematicians will arrive at the same result when simplifying an expression, thus maintaining consistency within the discipline.