Problem 53 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ 160-2 \cdot 5 \cdot 3^{2} $$

Step-by-Step Solution

Answer

The answer is 70.

Step 1: Evaluate the exponent

First, calculate the value of the exponent. Here, we have the term \(3^{2}\). \(3^{2} = 3 \times 3 = 9\)

Step 2: Multiply the results

Next, multiply the results with the remaining terms in the expression. The expression becomes: \(2 \times 5 \times 9\) First multiply \(2 \times 5 = 10\). Then multiply the result by 9, \(10 \times 9 = 90\).

Step 3: Subtract from initial value

Finally, subtract the result from the initial value in the expression: \(160 - 90 = 70\).

Key Concepts

Evaluating Expressions

Evaluating expressions involves performing operations in a specific order to reduce an expression to a single value. This process requires following the 'order of operations', often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In expressions without parentheses, start with exponents, then proceed to multiplication or division, and finally handle addition or subtraction. By following this order, you ensure the expression is simplified correctly.

Exponents

Exponents represent repeated multiplication of a number by itself. For example, in the exercise provided, we have the term \(3^2\). This means multiplying 3 by itself: \(3 \times 3 = 9\).

Exponents are a critical part of many mathematical expressions and must always be evaluated before any multiplications, divisions, additions, or subtractions. Understanding exponents' basics is key to correctly simplifying expressions.

Multiplication

Multiplication involves combining quantities by multiplying them. In the exercise, the expression \(2 \times 5 \times 9\) is evaluated by performing multiplications in sequence. First, you calculate \(2 \times 5 = 10\). Then multiply this result by 9 to get 90.

Always handle multiplications before moving on to additions or subtractions unless parentheses dictate otherwise. This ensures you maintain the correct order of operations.

Subtraction

Subtraction involves taking one quantity away from another. It is usually performed after all operations within parentheses, exponents, multiplications, and divisions are completed. In our exercise, the final step is to subtract 90 from 160: \(160 - 90 = 70\).

Remember to always follow the order of operations to ensure that the subtraction (and any other operations) are performed at the correct time in the sequence.