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Chapter 14: Problem 56

Evaluate. See Sections 1.7 and 7.7 $$ \frac{2\left(1-5^{3}\right)}{1-5} $$

Short Answer

Expert verified
The evaluated expression is 62.

Step by step solution

01

Simplify the Exponent

Begin by simplifying the expression inside the parentheses with the exponent. Calculate \(5^3\), which is \(5 \times 5 \times 5 = 125\). So, the expression becomes:\[2(1 - 125)\].
02

Evaluate the Parentheses

Now, evaluate the expression inside the parentheses: \(1 - 125\). This simplifies to \(-124\). Thus, the expression updates to \(2(-124)\).
03

Multiply the Coefficient

Multiply the coefficient 2 by the simplified expression inside the parentheses: \(2 \times -124 = -248\).
04

Simplify the Denominator

Evaluate the denominator: \(1 - 5\), which simplifies to \(-4\).
05

Divide the Result

Now divide the result from Step 3 by the simplified denominator from Step 4: \(-248 \div -4\). With two negative signs, the division yields a positive result: \(62\).

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Key Concepts

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

Simplifying Exponents
Exponents represent repeated multiplication of a number by itself. They are an efficient way to express large numbers. For instance, the expression \(5^3\) means that 5 is multiplied by itself twice more for a total of three times. Calculating this, we have:
  • \(5 \times 5 = 25\)
  • \(25 \times 5 = 125\)
Thus, \(5^3 = 125\). In algebraic expressions, simplifying exponents often comes first to break down the complexity of the terms involved. Always perform these calculations accurately to avoid errors in later steps.
Parentheses in Algebra
In algebra, parentheses are used to indicate which operations should be performed first. This is a key part of understanding order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When you see an expression like \((1 - 125)\), it means you must complete the subtraction before moving on to any outside calculations. Here:
  • \(1 - 125 = -124\)
Getting the calculation inside the parentheses correct is vital, as errors here can affect the entire solution.
Multiplying Coefficients
When we talk about multiplying coefficients in algebra, we're discussing the numbers that are placed in front of variables or expressions. The coefficient can be thought of as a multiplier. In the simplified expression \(2(-124)\), the number \(2\) is multiplied by \(-124\):
  • \(2 \times -124 = -248\)
Multiplying terms correctly is crucial, especially when dealing with negatives, as the sign of the result can change.
Division of Negatives
Dealing with negative numbers can initially seem tricky, but there's a consistent rule: dividing two negative numbers results in a positive number. This is because the negatives cancel each other out. In our problem, we have \(-248\) divided by \(-4\):
  • \(-248 \div -4 = 62\)
Two negative signs "flip" the result to positive. Being comfortable with this process is crucial, as it allows you to handle negative integers effectively in algebra and other mathematical contexts.

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Most popular questions from this chapter

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