Problem 36 - Algebra 1

Question

Evaluate the expression. $$\frac{1}{2} \cdot 26-3^{2}$$

Step-by-Step Solution

Answer

The evaluation of the expression \(\frac{1}{2} \cdot 26-3^{2}\) is 4.

Step 1: Solve the power operation

The given expression involves 3^2. The power operation means 3 should be multiplied by itself. So, 3^2 = 3 * 3 = 9.

Step 2: Perform the multiplication operation

Now, perform the multiplication operation in the expression. That is, multiply 1/2 by 26. This results in 1/2 * 26 = 13.

Step 3: Perform the subtraction operation

After performing the multiplication operation, the expression is '13 - 9'. Hence, the subtraction operation results in 13 - 9 = 4.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. Understanding this concept is crucial in solving equations efficiently. When you see a number raised to a power, like \(3^2\), it means you multiply the base (3) by itself the number of times indicated by the exponent (2). So, \(3^2\) is equal to \(3 \times 3\), which results in 9.

• The base is the number being multiplied.
• The exponent tells you how many times to multiply the base by itself.
• Exponents are sometimes called "powers."

Exponents are part of the "order of operations" in mathematics. This means they are calculated before multiplication, division, addition, and subtraction. Understanding this helps you solve equations effectively.

Multiplication

Multiplication is one of the basic operations in math used to find the total of one number added a certain number of times. In the context of solving expressions, multiplication needs to be performed after any present exponents and before any addition or subtraction.

In our example, we multiply \(\frac{1}{2}\) by 26. This involves multiplying each part of \(\frac{1}{2}\) by 26, which gives \(\frac{1 \cdot 26}{2 \cdot 1} = \frac{26}{2} = 13\).

• Use the commutative property: \(a \times b = b \times a\).
• The identity property of multiplication states \(a \times 1 = a\).
• Multiplication can be visualized using arrays or groups.

Understanding multiplication in expressions helps simplify problems before moving on to subtraction.

Subtraction

Subtraction is used to find the difference between numbers. In the "Order of Operations," it is performed after solving any multiplication, division, and exponentiation present.

In our expression, after calculating the multiplication to result in 13, we subtract 9 (from the exponential calculation earlier) from it. This means \(13 - 9\) gives us 4. Here's how subtraction fits into problem-solving:

• Subtraction can also be thought of as "finding the missing addend."
• It does not follow the commutative property: \(a - b eq b - a\).
• Understanding subtraction helps in balancing equations and solving for unknowns.

Bringing the pieces together in an expression means you subtract only after doing multiplications and working with exponents.