Problem 73 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Then simplify the answer. $$ \frac{8 \cdot 8}{10+3 \cdot 2} $$

Step-by-Step Solution

Answer

The simplified answer of the given expression is 4.

Step 1: Apply BODMAS Rule

First off, perform the operations in parentheses if there is any and in this case, there's none. Then, perform exponentiation operation, and here again, none is present in this expression. Next, perform multiplication or division operations from left to right. Here, we have \(8 \cdot 8\) in the numerator and \(10 + 3 \cdot 2\) in the denominator. We solve them like this: \(8 \cdot 8 = 64\) and \(10 + 3 \cdot 2 = 10 + 6 = 16\). The expression now becomes \(\frac{64}{16}\).

Step 2: Perform Division

Now, divide 64 by 16 to get the simplified answer. So, \(\frac{64}{16} = 4\).

Step 3: Verify the Answer

The final step is to verify if the calculated answer is integer and fully simplified, which indeed it's in this case.

Key Concepts

BODMAS Rule

Understanding the BODMAS Rule is crucial when solving algebraic expressions. BODMAS stands for Brackets, Order (or Exponents), Division, Multiplication, Addition, and Subtraction. This rule helps in deciding the order in which operations should be performed in a given mathematical expression.
- First, solve operations within brackets - Then move to exponents, like squares or square roots- Next, perform division and multiplication from left to right- Lastly, tackle addition and subtraction from left to right
In our exercise, there were no brackets or exponents, so we proceed directly to multiplication and addition. The multiplication in the numerator, \[8 \cdot 8 = 64\], was done first. In the denominator, the addition \[10 + 6 = 16\] was performed after finding \[3 \cdot 2\]. By following BODMAS, we get to an accurate solution.

Simplification

Simplification involves reducing an expression to its simplest form. This often means combining like terms, canceling factors when possible, or performing arithmetic operations like division or multiplication.

• Consider every term and operation in the mathematical expression.
• Apply arithmetic rules and the BODMAS rule to each part sequentially.
• Ensure to remember past steps, as omitting them may lead to errors.

The step-by-step method allows for correctly simplifying the expression. In our exercise, simplifying was necessary after evaluating: \[\frac{64}{16}\]. This involves performing a straightforward division to reach the simplest form, which is \[4\]. This entire process ensures the expression is as streamlined as possible.

Division in Algebra

Division in algebra involves distributing a numerator across a denominator or simplifying fractions. A firm grasp of division is vital for simplifying expressions correctly.
Here’s how to effectively tackle division in algebra:

• Ensure that you have simplified the numerator and denominator separately, considering all operations involved.
• Perform the division directly if the expression allows, as seen in our case with \[\frac{64}{16}\].
• Divide the two numbers to get a whole number if possible, thereby simplifying the expression entirely.

In our exercise, dividing 64 by 16 yielded 4. This uncomplicated solution confirmed that no further division or simplification was possible, indicating the expression was fully resolved.