Problem 94 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{-20+(-4)^{2}(3)}{1-5}\)

Step-by-Step Solution

Answer

The evaluated expression is -7.

Step 1: Evaluate the Exponentiation

The first operation in the expression is the exponentiation. Calculate \((-4)^2 = 16\) as squaring a negative number results in a positive number.

Step 2: Multiply the Results

Next, we take the result of the exponentiation (16) and multiply it by 3. So, \(16 \times 3 = 48\).

Step 3: Subtract in the Numerator

Replace the exponentiation and multiplication results into the expression. The expression then becomes \(-20 + 48\). Calculate \(-20 + 48 = 28\).

Step 4: Evaluate the Denominator

Simplify the denominator expression, \(1 - 5\). Calculate \(1 - 5 = -4\).

Step 5: Divide the Results

Now divide the result from the numerator by the result of the denominator. That is, \(\frac{28}{-4} = -7\).

Key Concepts

Exponentiation

Exponentiation involves raising a number, known as the base, to the power of another number, called the exponent. It's a way of multiplying a number by itself a certain number of times. For example, in the expression \((-4)^2\), the base is \(-4\), and the exponent is \(2\). This means you multiply \(-4\) by itself: \(-4 imes -4\).

• When squaring negative numbers like in our example, the result is positive. This is because multiplying two negative numbers results in a positive product.
• It's important to note the placement of the negative sign. \((-4)^2\) is different from \-4^2\, the latter is equivalent to \-(4^2)\.
• In our original exercise, \((-4)^2 \) equates to \(16\).

Understanding exponentiation helps in simplifying expressions efficiently and correctly.

Order of Operations

When solving algebraic expressions, it's crucial to follow a specific sequence to ensure accuracy. This sequence is known as the Order of Operations, often remembered by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• First, handle any operations inside parentheses or brackets. In this problem, we solve the exponent first since it is a part of the question segment \( (-4)^2 \).
• Next, address any exponents. We've calculated \((-4)^2 \) and moved on to multiplication, following the order.
• Multiplication and Division are performed as they appear from left to right. In our expression, we multiplied the result of the exponentiation by 3.
• Lastly, perform addition or subtraction from left to right. After calculating \(-20 + 48\), we simplified the denominator and completed the division.

A firm grasp of this order ensures that arithmetic operations are performed correctly.

Negative Numbers

Negative numbers can sometimes cause confusion, especially when paired with operations like multiplication, division, or exponentiation. It's important to remember a few key rules when working with them:

• Adding a negative number is essentially the same as subtraction. For instance, \(-20 + 48\) becomes \(-20 + 48 = 28\).
• When you multiply or divide two negative numbers, the result is positive. This is why \((-4)^2\) becomes \(16\), a positive number.
• However, if you multiply or divide a positive and a negative number, the result is negative. \(rac{28}{-4}\) results in \(-7\).

Grasping these concepts can help demystify the operations involving negative numbers and lead to accurate calculations.