/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Evaluate. \(\frac{3\left(1-2^{... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 55

Evaluate. \(\frac{3\left(1-2^{4}\right)}{1-2}\)

Short Answer

Expert verified
The result is 45.

Step by step solution

01

Evaluate the Power

Calculate the power of 2 raised to 4, which is written as \(2^4\). This equals: \(2 \times 2 \times 2 \times 2 = 16\).
02

Simplify the Expression in the Numerator

Substitute the result from Step 1 into the expression \(1 - 2^4\) in the numerator to get \(1 - 16\). This equals: \(1 - 16 = -15\).
03

Multiply the Results in the Numerator

Multiply the number in the numerator by 3: \(3 \times (-15) = -45\).
04

Simplify the Denominator

Evaluate the expression in the denominator: \(1 - 2 = -1\).
05

Divide the Numerator by the Denominator

Calculate the division of the evaluated numerator by the evaluated denominator: \(\frac{-45}{-1} = 45\).

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Key Concepts

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

Exponentiation
Exponentiation is a key concept in algebra that involves raising a number to a power. In the expression \(2^4\), the number 2 is referred to as the "base," and 4 is the "exponent." This tells us how many times we need to multiply the base by itself. For \(2^4\), we perform the following calculation:
  • Multiply 2 by itself four times: \(2 \times 2 \times 2 \times 2 = 16\).
Exponentiation is an efficient way to simplify calculations where large numbers are involved. This concept is foundational in algebraic expressions and helps in reducing complex problems into more manageable steps. Understanding how to evaluate exponents is crucial for working with polynomial equations, growth models, and various scientific calculations.
Numerical Evaluation
Numerical evaluation refers to replacing algebraic expressions with their numerical values. In the problem presented, after calculating \(2^4 = 16\), we substitute the value back into the expression. The component \(1 - 2^4\) becomes \(1 - 16\). By carrying out this substitution, we simplify the expression step by step:
  • First, \(1 - 16\) evaluates to \(-15\).
  • Next, multiplying by 3 gives us \(3 \times (-15) = -45\).
Understanding how to substitute and simplify numeric values is crucial as it breaks down a potentially complex expression into smaller, easier parts. It allows us to solve the expression methodically by reducing it to simple arithmetic operations.
Division of Integers
Division of integers involves dividing one integer by another. In algebra, this is often the final step in simplifying an expression. Our example was \( \frac{3(1-2^4)}{1-2}\).Once the numerator \(3 \times (-15) = -45\) and the denominator \(1 - 2 = -1\) are simplified, division takes place:
  • Divide \(-45\) by \(-1\) resulting in 45.
The division of integers follows straightforward rules:
  • If both integers have the same sign, the quotient is positive.
  • If the integers have different signs, the quotient is negative.
These simple rules help tackle a wide array of algebraic problems, making integer division an essential aspect of arithmetic and algebra.

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