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Magazine Chapter 1: Problem 93
Evaluate each expression. \(\frac{22+(3)(-2)^{2}}{-5-2}\)
Short Answer
Expert verified
The expression evaluates to \(-\frac{34}{7}\).
Step by step solution
01
Evaluate Exponents
Start by evaluating the exponent in the expression. Here, \((-2)^2\) needs to be calculated first. Squaring \(-2\) gives \((-2) \times (-2) = 4\).
02
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 \((3)(4) = 12\).
03
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 \(22 + 12 = 34\).
04
Simplify the Denominator
Now simplify the denominator, which is \(-5 - 2\). Subtract 2 from -5 to get \(-5 - 2 = -7\).
05
Division
Divide the simplified numerator by the simplified denominator. The division is \(\frac{34}{-7}\).
06
Final Result
Simplify the division step if necessary. Here, \(\frac{34}{-7}\) is already in its simplest form, yielding a final result of \(-\frac{34}{7}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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, \( 2^{-1} = \frac{1}{2} \).
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鈥檚 what you should always remember:
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鈥檚 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.
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:
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鈥檙e in their simplest possible form.
