Problem 13 - Elementary Algebra

Question

Write the principal and secondary square roots of each number. Use a calculator to obtain a decimal approximation for the two square roots of 35 . Round to two decimal places.

Step-by-Step Solution

Answer

Answer: The principal square root of 35 is approximately 5.92, and the secondary square root is approximately -5.92.

Step 1: Find the principal square root of the number

To find the principal square root of 35, use the square root symbol (√) followed by the number: Principal square root = √35

Step 2: Use a calculator to find the decimal approximation of the principal square root

Using a calculator, find the decimal approximation of √35 up to two decimal places: Principal square root ≈ 5.92

Step 3: Find the secondary square root of the number

The secondary square root is the negative counterpart of the principal square root: Secondary square root = -√35

Step 4: Use a calculator to find the decimal approximation of the secondary square root

Using a calculator, find the decimal approximation of -√35 up to two decimal places: Secondary square root ≈ -5.92 Thus, the principal and secondary square roots of 35 are approximately 5.92 and -5.92, respectively.

Key Concepts

Decimal Approximation

Decimal approximation is the process of finding a decimal number that is close enough to a given value to be useful for practical purposes. In our context, it is used to express square roots, which are often irrational numbers, in a simplified decimal form.
When calculating the square root of 35, we find it to be an irrational number because its exact value cannot be exactly expressed as a finite or repeating decimal.
By using a calculator, we can compute a precise decimal approximation of the square root of 35, which is around 5.92 when rounded to two decimal places.
This approximation allows us to handle the number in calculations more smoothly, without the need for complex fractions or continued radicals. With decimal approximation, remember to round off to the required number of decimal places—in our case, the second decimal place.

Principal Square Root

The principal square root refers to the positive version of the square root of a number. For any non-negative number \( x \), the principal square root is denoted by \( \sqrt{x} \).
For the number 35, the principal square root is \( \sqrt{35} \). Calculators and standard functions typically focus on this root, providing a positive result.
The principal square root is essential because it represents the non-negative solution to the equation \( x^2 = a \), where \( a \) is the number under the root sign.
In practical terms, finding \( \sqrt{35} \) using a calculator gave us a decimal approximation of approximately 5.92.

• This positive value is what we use when determining the principal square root of a number.

It establishes a standard go-to root that is widely recognized and trusted in mathematical operations and solutions.

Secondary Square Root

The secondary square root is the negative counterpart of the principal square root. While the principal deals with the positive solution, the secondary tackles the negative one.
For a number like 35, where the principal square root is \( \sqrt{35} \approx 5.92 \), the secondary square root is \(-\sqrt{35} \approx -5.92\).
This aspect of square roots is crucial because every positive number actually has two square roots: one positive (principal) and one negative (secondary).
When solving equations involving square roots, it is vital to consider both roots for completeness.

• In our exercise, just like we found the decimal approximation of the principal square root using a calculator, the same was done for the secondary root.
• The secondary value of -5.92 helps ensure that both roots are accounted for.

Always remember that acknowledging the secondary square root ensures all possible solutions are explored in mathematical problems.