Problem 81 - Elementary Algebra

Question

Simplify \(\frac{4\left(7^{2}-6 \cdot 2^{3}\right)}{2^{2}}\).

Step-by-Step Solution

Answer

Question: Calculate and simplify the following expression: \(\frac{4[7^{2} - (6 \cdot 2^{3})]}{2^{2}}\). Answer: The simplified expression is 1.

Step 1: Calculate the values of \(7^{2}\) and \(6 \cdot 2^{3}\)

Calculate the value of \(7^{2}\): \(7^{2} = 7 \cdot 7 = 49\). Calculate the value of \(6 \cdot 2^{3}\): \(2^{3} = 2 \cdot 2 \cdot 2 = 8\) and then multiply by 6: \(6 \cdot 8 = 48\).

Step 2: Subtract the second value from the first value

Subtract the value of \(6 \cdot 2^{3}\) from the value of \(7^{2}\): \(49 - 48 = 1\).

Step 3: Multiply the result by 4

Multiply the result obtained in step 2 by 4: \(4 \cdot 1 = 4\).

Step 4: Divide the result by \(2^{2}\)

Calculate the value of \(2^{2}\): \(2^{2} = 2 \cdot 2 = 4\). Divide the result obtained in step 3 by the value of \(2^{2}\): \(\frac{4}{4} = 1\). The simplified expression is 1.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. When you see something like \(7^2\), it means \(7\) is multiplied by itself, giving you \(7 \times 7 = 49\).

Here are a few essential points to remember about exponents:

• An exponent tells you how many times the base number is used in a multiplication.
• If the exponent is \(2\), this specific operation is called "squaring" a number.
• Negative exponents represent the reciprocal of the base raised to the opposite positive power, while a zero exponent means that any non-zero base raised to the power of zero equals one.

In the original exercise, exponents are used with both \(7\) and \(2\). Evaluating \(7^2\) shows how powerful exponents can be in calculations.
By simplifying step by step, complex expressions involving exponents become more manageable, which is a vital skill in algebra.

Order of Operations

When simplifying expressions in algebra, the order in which you perform operations is crucial. This precise structure prevents ambiguity and ensures consistency across mathematical calculations.

A common acronym to remember is PEMDAS, which stands for:

• Parentheses: Solve expressions within parentheses first.
• Exponents: Next, compute powers and roots.
• Multiplication and Division: Perform these operations as they appear from left to right.
• Addition and Subtraction: Finally, handle these operations from left to right.

In the original expression, the order of operations is followed diligently:

First, calculate the exponents \((7^2 \text{ and } 2^3)\). Then, the multiplication (\(6 \times 2^3\)). After resolving these, you subtract, multiply by \(4\), and finally divide by \(2^2\).

Mastery of the order of operations helps students systematically break down and simplify complex numerical expressions without making errors.

Numerical Expressions

Numerical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division. They sometimes also include exponents, parentheses, or other mathematical symbols.

Simplifying numerical expressions often requires following the order of operations to avoid mistakes.

The practice of simplifying expressions is an opportunity to exercise arithmetic and algebraic skills, letting students:

• Understand mathematical relationships between numbers.
• Combine different operations within a single set of numbers.
• Manipulate parts of an equation in a logical order.

Becoming confident in working with numerical expressions lays the groundwork for understanding more complex algebraic concepts. Each solved example, like the one involving \(\frac{4(7^{2}-6 \cdot 2^{3})}{2^{2}}\), reinforces these foundational skills, making future encounters with multi-step problems more intuitive.

It's all about piecing each part of the puzzle together correctly to get to the simplest form or the exact value.