Problem 101 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ -\left|-5 \cdot 7^{2}\right|-30 $$

Step-by-Step Solution

Answer

The expression evaluates to -275.

Step 1: Evaluate the Exponent

First, calculate the square of 7. The expression inside the absolute value contains the term \(7^2\), which means \(7\) is multiplied by itself: \(7 \times 7 = 49\).

Step 2: Multiply the Terms Inside the Absolute Value

Now, multiply the result from step 1 by -5, as indicated in the expression \(-5 \cdot 49\). The calculation is \(-5 \times 49 = -245\).

Step 3: Apply the Absolute Value

Next, apply the absolute value to \(-245\). The absolute value of a number is always non-negative, so \(|-245| = 245\).

Step 4: Apply the Negative Sign Outside the Absolute Value

The original expression has a negative sign outside the absolute value. We need to put the negative sign back on \(245\), turning it into \(-245\).

Step 5: Subtract 30

Finally, subtract 30 from \(-245\): \(-245 - 30 = -275\).

Key Concepts

Absolute Value

The absolute value is a fundamental concept in algebra that indicates the distance of a number from zero on the number line, without considering the direction. It's represented with vertical bars around the number, like \( |x| \). The outcome is always non-negative, whether the number inside the bars is positive or negative.

For instance, \( |-245| = 245 \). This is because distance cannot be negative. When evaluating an expression with absolute value, remember:

• If the number is negative, just remove the negative sign.
• If the number is positive, it remains unchanged.
• If the number is zero, the absolute value is also zero.

Applying this to our exercise, converting \( |-245| \) to \( 245 \) was necessary because we are dealing with distances, not directions.

Exponents

Exponents are a way to express repeated multiplication of the same number by itself. It is shown as a small number, called the power, placed to the upper right of a base number. For example, in \( 7^2 \), 7 is the base and 2 is the exponent or power.

This notation indicates that you multiply the base number by itself as many times as the exponent specifies. So, \( 7^2 \) means \( 7 imes 7 = 49 \). This step is fundamental in simplifying an expression, as seen in the initial stage of our problem.

• The base is the number that is being multiplied.
• The exponent tells us how many times we multiply the base by itself.
• Exponents can be any whole number and are crucial for compactly expressing large products.

Understanding and evaluating exponents correctly helps make complex calculations manageable and is often one of the first steps in algebraic evaluations.

Order of Operations

In algebra, the order of operations is a set of rules that define the correct sequence to evaluate a mathematical expression. It's essential to follow these rules to get the intended result.

Usually remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), these rules tell us:

• First, perform calculations inside Parentheses.
• Next, evaluate any Exponents.
• Then, complete any Multiplication or Division, from left to right.
• Lastly, perform Addition or Subtraction, from left to right.

In our exercise, we first handled the exponent by calculating \( 7^2 \), then multiplied the result by -5, wrapped it in absolute value, and finally applied all signs as stated in the order to reach the correct solution. Applying the order of operations ensures that everyone interprets and simplifies mathematical expressions the same way.