Problem 3 - Prealgebra

Question

Find each of the following square roots without using a calculator. $$\sqrt{81}$$

Step-by-Step Solution

Answer

The square root of 81 is 9.

Step 1: Understand the Problem

We need to find the square root of 81, which is the number that when multiplied by itself gives 81.

Step 2: Identify Perfect Squares

Recall perfect squares are numbers whose square roots are integers. Examples include 1, 4, 9, 16, 25, and so on. We need to determine if 81 is one of these.

Step 3: Estimate and Check

Find a number that, when multiplied by itself (squared), equals 81. Since 9 times 9 equals 81, we can say that the square root of 81 is 9. \[9 \times 9 = 81\]

Step 4: Verify Solution

Verify by recalculating: \[9^2 = 9 \times 9 = 81\] Since this is correct, the square root of 81 is confirmed to be 9.

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers that are the product of an integer multiplied by itself. In simpler terms, if you can take an integer and multiply it by itself to get a number, then that number is a perfect square. For instance:

• 1 is a perfect square because \(1 \times 1 = 1\).
• 4 is a perfect square because \(2 \times 2 = 4\).
• 9 is a perfect square because \(3 \times 3 = 9\).
• 16 is a perfect square because \(4 \times 4 = 16\).
• 25 is a perfect square because \(5 \times 5 = 25\).

These are just some of the lower perfect squares, but they continue infinitely as you go higher in numbers. Recognizing perfect squares helps when calculating square roots because, if a number is a perfect square, its square root will be an integer. This is particularly useful in mathematical problems, like the one given, where you need to find the square root without a calculator.

Estimating Square Roots

Estimating square roots is especially useful when dealing with numbers that aren't perfect squares. While the process is straightforward with perfect squares, estimating allows you to find approximate values otherwise. For example, if you're looking for the square root of a number like 50, which isn’t a perfect square:

• First, determine two consecutive perfect squares the number 50 sits between. In this case, it’s between 49 and 64 since \(7 \times 7 = 49\) and \(8 \times 8 = 64\).
• This gives an estimate: the square root of 50 is slightly more than 7 but less than 8.

Once this range is identified, you can refine your estimate. This process involves testing decimal values such as 7.1 or 7.2 to approximate further. By trying out these decimals, you’ll get closer and closer to the actual square root, facilitating a better understanding of values that lie between perfect squares.

Exploring Integer Roots

Integer roots refer to numbers that, when squared, result in perfect squares. As mentioned before, perfect squares like 1, 4, and 9 correspond to integer roots of 1, 2, and 3, respectively. This concept is crucial, because it demonstrates the relationship between numbers and their square roots clearly.
In the given exercise where the task was to find \(\sqrt{81}\), we identify 9 as the integer root. Here's why:

• The number 9, when squared (\(9 \times 9\)), results in 81, proving that 81 is a perfect square.
• Thus, the square root of 81 is an integer, specifically 9, which makes it also the integer root of this perfect square.

Focusing on integer roots enables an easier computation of square roots, especially in cases where the numbers involved are neatly perfect squares. Grasping integer roots helps simplify tasks that deal with rooted figures in mathematics, making problem-solving a more intuitive process.