Problem 47 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ (2 \cdot 7)^{-1} $$

Step-by-Step Solution

Answer

The evaluated expression rounded to the nearest ten-thousandth is \(0.0714\).

Step 1: Carry out the multiplication

We first carry out the multiplication in the parentheses: \(2 \cdot 7 = 14\)

Step 2: Take the inverse

Then we take the inverse of 14. The inverse of a number is given by: \(1 / \text{number}\). So \(14^{-1} = 1 / 14\)

Step 3: Calculate the value

On calculating, we get a decimal number: Approximately \(0.07142857142857142\)

Step 4: Round the answer to the nearest ten-thousandth

We then round this answer to the nearest ten-thousandth. Ten-thousandth place is the fourth place after decimal. So, our answer becomes \(0.0714\) after rounding.

Key Concepts

Understanding Inverse Operations

Inverse operations are like magic tricks in math: they help you undo things! If you multiply a number, its inverse will divide it back to 1. Easy-peasy, right? The inverse of a number is essentially 1 divided by that number, written as \(x^{-1} = \frac{1}{x}\).

• If you love cupcakes, think of it this way: you have 14 cupcakes (the number), and eating cupcakes (inverse operation) gives you one whole cake, represented as \(\frac{1}{14}\).
• In our problem, we first needed to multiply numbers (2 and 7), and then find the inverse to simplify to just one cupcake!

Understanding inverse operations helps you switch numbers and operations around, bringing you back to where you started, or in this case, to a simple solution!

Mastering Multiplication

Multiplication is the mathematical operation where numbers are combined in groups. Think of it like repeatedly adding a number. For instance, 2 multiplied by 7 is the same as adding 2 together 7 times, resulting in 14. Pretty neat!
But why is multiplication so cool?

• It transforms complex problems into simpler ones. Imagine repeatedly having to add numbers to reach a sum. Multiplication does that in a snap!
• It lays the groundwork for more advanced math concepts like fractions, which leads us to understanding operations like finding an inverse.

By mastering multiplication, you're not just juggling numbers; you're setting the stage for diving deeper into math magic, understanding the whole picture, using tricks like inverses easily.

Precision with Decimal Places

Decimals are all about precision. They're like zooming in on a mathematical map to get details to the most accurate point you need. Decimal places ensure that we’re as accurate as needed without overwhelming with unnecessary details.

• Every spot after the decimal point reveals just how precise you'd like to be. The first spot is the tenths, next are hundredths, then thousandths, and finally in our case, the ten-thousandths.
• Rounding numbers is like tidying up the room: keeping what's important and letting go of the unimportant so your answer looks neat and accurate.

In the given exercise, our final task was to round the answer \(0.07142857142857142\) to the nearest ten-thousandth. By doing this, you simplify your solution to \(0.0714\), making it easier to handle while still being precise enough for most needs.