Problem 107 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{-5^{2} \cdot 10+5 \cdot 2^{5}}{-5-3-1} $$

Step-by-Step Solution

Answer

The value of the expression is 10.

Step 1: Evaluate the Powers

First, evaluate the powers in the expression: \(-5^2\) gives you \(25\), because the exponent applies only to the 5. \(2^5\) gives you \(32\).

Step 2: Multiply and Simplify the Numerator

Now, evaluate the multiplication and addition in the numerator: \(25 \cdot 10 = 250\) and \(5 \cdot 32 = 160\). Then, compute the entire numerator: \(-250 + 160 = -90\).

Step 3: Simplify the Denominator

Simplify the denominator:\(-5 - 3 - 1\) gives you \(-9\).

Step 4: Divide the Numerator by the Denominator

Finally, divide the simplified numerator by the denominator: \(-90 \div -9 = 10\).

Key Concepts

Order of Operations

Order of operations is a fundamental concept in algebra that helps us solve expressions systematically. To avoid confusion and ensure consistency, we follow a specific order, known as PEMDAS:

• Parentheses
• Exponents (including roots, like square roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, the expression \[\frac{-5^{2} \cdot 10+5 \cdot 2^{5}}{-5-3-1}\]requires us to strictly follow this order. By addressing the exponents first, then multiplication and division, and finally addition and subtraction, we ensure that the complex calculations become manageable. This method helps prevent common errors in solving algebraic expressions.

Exponentiation

Exponentiation is an important operation in which a number (the base) is multiplied by itself a number of times (the exponent). It’s crucial to note that the exponent only affects the base right next to it, unless parentheses specify otherwise.
For example, in \(-5^2\), the exponent \(2\) applies only to \(5\), giving us \(25\), not \(-25\). Similarly, \(2^5\) means \(2 \times 2 \times 2 \times 2 \times 2 = 32\).
Understanding how to correctly apply exponents is crucial for evaluating algebraic expressions correctly and efficiently. Mistakes in applying exponents often lead to incorrect answers, so it's a key concept to master.

Multiplication and Division

In algebraic expressions, multiplication and division are performed after exponents but before addition and subtraction. These operations are carried out from left to right.
In our exercise, once the powers were evaluated, the next step was to handle multiplication and division. For instance:

• \(25 \cdot 10 = 250\)
• \(5 \cdot 32 = 160\)

The resulting products then assist in computing the numerator. Once the numerator (\(-250 + 160 = -90\)) and denominator (\(-9\)) are simplified, division is used to find the final result of the expression. This systematic approach ensures accuracy and clarity in solving complex algebraic expressions.