Question

Find each square root. Round to the nearest tenth, if necessary. \(\sqrt{126}\)

Short answer

The square root of 126, rounded to the nearest tenth, is 11.2.

Step-by-step solution

Understand the Problem

We are asked to find the square root of 126 and round it to the nearest tenth.

Estimate the Square Root

Recognize that 126 falls between the perfect squares 121 (which is 11²) and 144 (which is 12²). So, the square root of 126 is between 11 and 12.

Calculate the Square Root

Use a calculator to find the approximate value of the square root of 126. You should find that \( \sqrt{126} \approx 11.22497216 \).

Round to the Nearest Tenth

Look at the hundredths place to decide how to round. Since the hundredths digit is 2, we round down. So, \( 11.22497216 \) rounded to the nearest tenth is 11.2.

Key concepts

Understanding Rounding

Rounding is a mathematical technique used to reduce the number of digits in a number while keeping its value close to the original. In our exercise, we need to round the square root of 126 to the nearest tenth. This means we want to keep one decimal place. After calculating the square root, we get approximately 11.22497216.
To determine how to round, we look at the digit in the hundredths place, which is the second digit to the right of the decimal point. Here, that's the number 2. If this digit is less than 5, we round down, meaning we keep the first decimal digit as it is. If it were 5 or more, we'd round up, adding 1 to the tenth's place. In our case, the number is less than 5, so we round 11.22497216 to 11.2.
Rounding helps simplify calculations and make numbers easier to work with, especially when exact precision is not necessary.

How to Estimate the Square Root

Estimating is a crucial skill when dealing with square roots, particularly if you're aiming to get a quick idea of what a more complex calculation may yield. When estimating square roots, the goal is to find which two perfect squares the number lies between.
Perfect squares are numbers like 1, 4, 9, 16, and so on, which result from multiplying an integer by itself. In our scenario, 126 is sandwiched between 121 and 144, the perfect squares of 11 and 12, respectively. This tells us that the square root of 126 is somewhere between 11 and 12.
Knowing these bounds helps adjust our expectations for the calculated value and ensures we double-check our math if we get a result that seems out of place. It's a great way to develop intuition around numbers and calculations.

Approximating Complex Calculations

Approximation allows us to get 'close enough' answers without having to deal with overly complex maths. For square roots, this often involves using a calculator for precision. Once we've estimated that the square root of 126 is between 11 and 12, we specifically find it to be approximately 11.22497216 with a calculator.
This process of approximation helps us handle situations where exact numbers are either impractical or unnecessary. It also forms the backbone of many advanced mathematical practices where precise calculations require significant time or resources that aren't always available.
Approximations let us work with numbers dynamically but still maintain a high degree of accuracy, thereby making mathematical operations more accessible and manageable in everyday situations.