Problem 93 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{22+(3)(-2)^{2}}{-5-2}\)

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{34}{7}\).

Step 1: Evaluate Exponents

Start by evaluating the exponent in the expression. Here, \((-2)^2\) needs to be calculated first. Squaring \(-2\) gives \((-2) \times (-2) = 4\).

Step 2: Calculate Products

Next, calculate the product \((3)(-2)^2\). After evaluating the exponent in Step 1, replace \((-2)^2\) with 4, which makes the expression \((3)(4)\). Compute the product, obtaining \((3)(4) = 12\).

Step 3: Simplify the Numerator

Replace the exponentiated and multiplied part in the numerator with 12 to simplify as much as possible. Add it to 22 to get \(22 + 12 = 34\).

Step 4: Simplify the Denominator

Now simplify the denominator, which is \(-5 - 2\). Subtract 2 from -5 to get \(-5 - 2 = -7\).

Step 5: Division

Divide the simplified numerator by the simplified denominator. The division is \(\frac{34}{-7}\).

Step 6: Final Result

Simplify the division step if necessary. Here, \(\frac{34}{-7}\) is already in its simplest form, yielding a final result of \(-\frac{34}{7}\).

Key Concepts

Exponentiation

Exponentiation is the process of raising a number to a specified power. It involves multiplying a number by itself a certain number of times. For example, \((-2)^2\) means you multiply \(-2\) by itself:\((-2) \times (-2) = 4\).
This tells you that the square of a negative number results in a positive product if the exponent is even, because multiplying two negative numbers gives a positive result.
Here's how exponentiation works for different exponents:

• \( a^1 = a \:\) any number to the power of 1 is itself.
• \( a^0 = 1 \:\) any non-zero number to the power of 0 is 1.
• With a positive exponent: for example, \( 3^2 = 9 \).
• With a negative exponent: for instance, \( 2^{-1} = \frac{1}{2} \).

Understanding exponentiation is key in simplifying expressions, especially when dealing with polynomials and equations.

Numerators and Denominators

In fraction notation, the numerator and denominator have specific roles. The numerator is the top number, and it represents how many parts of a whole we have. Conversely, the denominator, which is the bottom part, indicates how many equal parts the whole is divided into.
For instance, in the fraction \(\frac{22+(3)(4)}{-5-2}\), the numerator is \(22 + (3)(4)\), which simplifies from the previous operation steps, while the denominator is \(-5 - 2\).
Here’s what you should always remember:

• The numerator can be a sum, a product, or a combination of both. It's the "top number".
• The denominator dictates the size of each segment of the whole. If you change it, you're essentially changing the partitioning of the whole.
• To simplify a fraction, first simplify both the numerator and the denominator separately.

This method of handling numerators and denominators helps in accurately simplifying expressions and arriving at the correct answer.

Division of Integers

Division of integers refers to distributing a dividend by a divisor. It may result in a whole number or a fraction, depending on whether the dividend is completely divisible by the divisor.
In the expression \(\frac{34}{-7}\), the division is straightforward: It involves putting the numerator 34 over the denominator -7. The result is a fraction, \(-\frac{34}{7}\), which indicates how many times -7 fits into 34.
Key points on dividing integers include:

• When both the dividend and divisor are positive or both are negative, the result is positive.
• If one is negative and the other is positive, the result is negative.
• Simplification may not reduce fractions composed of integers into a whole number, but ensures they’re in their simplest possible form.

Understanding the division of integers is fundamental to solving algebraic expressions, allowing you to transition smoothly between different forms of representing numbers.