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Chapter 7: Problem 59

Simplify each of the following as completely as possible. $$\frac{-2^{4}+3^{2}}{(-4+3)^{2}}$$

Short Answer

Expert verified
-7.

Step by step solution

01

- Evaluate the Numerator

Calculate the values in the numerator: First, simplify finally it in the step by in -4 power 2 plus brackers simplifiy containing 3 multiplies numerator top count, done
02

- Evaluate the Denominator

Calculate the values in the denominator: Simplify (-4+3) inside the parentheses first: Now the denmoniator
03

Combine values together

Place evaluated in numerator and denominator together, respectively. final combine

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluate Expressions
Evaluating expressions involves simplifying them step by step. It means solving for the value of the expression.
If an expression has variables, you may need to substitute numbers for those variables first before simplifying. However, in this example, we only deal with numbers.

To evaluate an expression:
  • Take note of the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)
  • Simplify each part of the expression according to these rules.
For the exercise, you need to simplify both the numerator and the denominator before combining them.
Numerator and Denominator
In fractions, the numerator is the top part, and the denominator is the bottom part. Both parts must be simplified separately before combining them.

  • For the numerator \( (-2^4 + 3^2) \), solve each part individually. Calculate \( -2^4 = -(2*2*2*2) = -16 \) and \(3^2 = 3*3 = 9 \). Then sum them up: \( -16 + 9 = -7 \).
  • For the denominator (\( (-4 + 3)^2 \)), simplify inside the parentheses first: \( -4 + 3 = -1 \), and then square the result: \( (-1)^2 = 1 \).
Finally, combine the evaluated numerator and denominator to get the simplified fraction. Here, we get \( \frac{-7}{1} = -7 \).
Order of Operations
The order of operations is crucial in simplifying mathematical expressions correctly. Using the acronym PEMDAS helps remember the order: Parentheses first, then Exponents, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).

For the exercise provided:
  • Start with parentheses: \( (-4 + 3) \) simplifies to \( -1 \).
  • Next, evaluate exponents: compute \( -2^4 = -16 \) and \( 3^2 = 9 \).
  • Then add the values obtained for the numerator: \( -16 + 9 = -7 \).
  • Finally, simplify the denominator by evaluating the exponent: \( (-1)^2 = 1 \).
Combining these results, you achieve the final value of the simplified fraction, \( -7 \).

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