Problem 62 - Elementary and Intermediate Algebra

Question

Simplify using a calculator. Round your answer to the nearest thousandth. $$ \frac{46-(3-8)^{3}}{2\left[35-(18-26)^{2}\right]} $$

Step-by-Step Solution

Answer

-2.948

Step 1 - Simplify the Expression Inside the Parentheses

First, simplify the expressions inside the parentheses. Calculate \( (3-8) \rightarrow -5 \and\ (18-26) \rightarrow -8 \)

Step 2 - Calculate the Exponents

Next, calculate the exponents for the values we derived in Step 1:\[(-5)^3 = -125 \]and\[(-8)^2 = 64\]

Step 3 - Substitute Back and Simplify

Substitute these values back into the original equation:\[\frac{46 - (-125)}{2 \times (35 - 64)}\]

Step 4 - Simplify the Numerator

Simplify the numerator:\[46 - (-125) = 46 + 125 = 171\]

Step 5 - Simplify the Denominator

Simplify the denominator:\[35 - 64 = -29\]Then multiply by 2:\[2 \times (-29) = -58\]

Step 6 - Simplify the Fraction

Divide the simplified numerator by the denominator:\[\frac{171}{-58} \rightarrow -2.948\]

Key Concepts

arithmetic operations

Arithmetic operations are the basic building blocks of math. They include addition, subtraction, multiplication, and division. To simplify algebraic expressions, you must perform these operations in the correct order.
For example, in the given problem, we perform subtraction inside the parentheses first: \(3 - 8 = -5\) and \(18 - 26 = -8\). These operations are straightforward but essential for progressing in the problem.

order of operations

The order of operations dictates the sequence in which mathematical operations should be performed to correctly simplify expressions. This order can be remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In our problem, we start by simplifying inside the parentheses: \((3 - 8)\) and \((18 - 26)\). Next, we handle the exponents: \((-5)^3\) and \((-8)^2\). Following that, we perform the remaining arithmetic operations.

fractions

Fractions represent parts of a whole and consist of a numerator (top part) and a denominator (bottom part). Simplifying fractions often involves simplifying the numerator and denominator separately before dividing. In the given exercise, we have the fraction: \( \frac{46 - (-125)}{2 \times (35 - 64)} \).
We simplify the numerator: \(46 - (-125) = 171\), and the denominator: \(2 \times (35 - 64) = -58\). Finally, dividing the simplified numerator by the simplified denominator gives us the final answer.

exponents

Exponents represent repeated multiplication of a number by itself. In our problem, exponents are used in the expressions \((-5)^3\) and \((-8)^2\).
Calculating \((-5)^3\) involves multiplying -5 by itself three times: \(-5 \times -5 \times -5 = -125\).
Similarly, calculating \((-8)^2\) involves multiplying -8 by itself two times: \(-8 \times -8 = 64\). The results of these exponential calculations are then used to simplify the remaining steps in the problem.

using a calculator

Using a calculator can greatly simplify complex calculations, especially when dealing with large numbers or multiple steps. To ensure accuracy, follow these steps:

• First, perform operations inside parentheses.
• Next, calculate exponents.
• Then, perform multiplication and division.
• Finally, handle addition and subtraction.

In our problem, you would input each step sequentially: first solving \((3-8)\) and \((18-26)\), then calculating exponents, and so on. Carefully using a calculator helps ensure you arrive at the correct final answer.