Problem 53 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$ \sqrt{5} $$

Step-by-Step Solution

Answer

The value of \( \sqrt{5} \) to the nearest hundredth is 2.24.

Step 1: Evaluate the Square Root

The first step is to calculate the square root of 5. Using a calculator or by solving it manually, the result can be found, which is approximately 2.236067977.

Step 2: Round the Result

The result calculated in step 1 is a non-terminating number. So, the next step is to round the result to the nearest hundredth. This gives us 2.24.

Key Concepts

Approximating Square Roots

Finding the exact square root of a number is not always possible, especially for numbers that are not perfect squares.
In such cases, approximation becomes our best friend as it helps us arrive at a value that is close enough for practical use.
Approximating the square root involves a bit of calculation and sometimes the use of estimation techniques.
The square root of 5, for instance, is not a simple whole number.
Pursuing an approximation, the calculator or long division method can be used to find a more precise value.

• The calculator offers a fast approximation, displaying the square root of 5 as approximately 2.236.
• The long division method, although a bit tedious, provides accuracy through systematic calculations.

Approximations are essential tools when dealing with real numbers, proving useful across various applications in mathematics and science.

Rounding Numbers

When we compute numbers with many decimal places, rounding becomes essential to simplify our calculations.
The objective is to reduce the number while keeping it close to its original value.
Rounding follows specific rules depending on which decimal place you target.
For example, rounding the number from our earlier calculation of \(\sqrt{5}\) (approx. 2.236067977) to the nearest hundredth requires looking at the third decimal place.

• If that digit (in this case, 6) is 5 or more, you round up the preceding digit, increasing the second decimal place.
• If it were 4 or less, the second decimal place would remain the same.

Thus, rounding 2.236067977 to the nearest hundredth results in 2.24.
Rounding is greatly beneficial in making complex numbers easy to work with, while still maintaining a reasonable level of accuracy.

Exact Values

Exact values are numbers expressed without any approximation. They retain their full precision, usually expressed in fractional or whole number form.
For numbers that are perfect squares, calculating a square root results in an exact value.

• For instance, the square root of 4 is exactly 2, since 2 times 2 gives 4.
• Likewise, \(\sqrt{9}\) is exactly 3, because 3 times 3 is 9.

However, when dealing with numbers like 5, the exact square root cannot be neatly expressed in decimal or whole number form.
In these cases, we either accept the number as a square root symbol \(\sqrt{5}\), or provide an approximation as discussed.
Exact values are crucial in many fields, allowing for precise calculations without the loss of information through rounding or approximation.