Problem 16 - College Algebra

Question

Simplify the given expression. $$ 9-18 \div 3^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Evaluate the Exponent

Begin with evaluating the exponent in the expression. The given expression is \(9 - 18 \div 3^{2}\). Here, calculate \(3^2\) first:\[3^2 = 3 \times 3 = 9\]Now the expression becomes \(9 - 18 \div 9\).

Step 2: Perform Division

Next, perform the division part of the expression. You now have \(9 - 18 \div 9\). Divide 18 by 9:\[18 \div 9 = 2\]Thus, the expression simplifies to \(9 - 2\).

Step 3: Subtract the Result

Finally, subtract the result from Step 2 from the initial number. Now you just need to calculate \(9 - 2\):\[9 - 2 = 7\]This gives you the simplified form of the expression.

Key Concepts

Exponents in Algebra

Understanding exponents is a critical part of simplifying expressions in algebra. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(3^2\), 3 is the base and 2 is the exponent. This means you multiply 3 by itself: \(3 \times 3\), which equals 9.

Exponents make calculations more compact and efficient.

• For positive integers, an exponent shows repeated multiplication.
• For instance, \(5^3\) means \(5 \times 5 \times 5 = 125\).

Handling exponents correctly is crucial because they affect the outcome of the entire mathematical expression. When you see an expression in algebra with exponents, remember to address them first while simplifying.

Order of Operations

The order of operations is a fundamental concept that determines the sequence in which mathematical operations should be performed. It is often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This sequence ensures that everyone solves equations uniformly, arriving at the same result.

In the given expression \(9 - 18 \div 3^2\), the order of operations dictates that you:

• First calculate the exponent \(3^2\).
• Then perform the division \(18 \div 9\).
• Finally do the subtraction \(9 - 2\).

Following these steps in the correct order guarantees accurate simplification of expressions and helps in avoiding errors. Every operation has its place, and skipping these rules might change the meaning of the expression entirely.

Division in Algebra

Division in algebra can sometimes be a tricky operation, especially when mixed with other operations like exponents and subtraction. In our example, \(18 \div 3^2\) is a critical step to simplify the expression. Remember:

• Division reduces the magnitude of numbers by splitting the numerator into parts equal to the divisor.
• Here, it transforms 18 into 2 when divided by 9, as \(18 \div 9 = 2\).

When performing division:

• Focus on the pair of numbers involved.
• Ensure to complete any preceding operations dictated by the order of operations.

Being methodical and adhering to these principles helps in simplifying expressions without confusion or missteps.