Problem 54 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$169$$

Step-by-Step Solution

Answer

The square roots of 169 are 13 and -13.

Step 1: Finding the Square Roots

The first step is to find the square roots of 169 by using the square root operation. Doing so, we see that there are two answers: \( \sqrt{169} = 13 \) and \( -\sqrt{169} = -13 \). Both 13 and -13 are square roots because squaring either number results in 169.

Step 2: Checking the Results

The second step is to verify these solutions by squaring them. If we square 13, \( (13)^2 = 169 \), and if we square -13 as well, \( (-13)^2 = 169 \). Since squaring both -13 and 13 result in 169, we can be certain these are the correct square roots of 169.

Key Concepts

Square Root Operation

Understanding the square root operation is crucial for solving many mathematical problems. It essentially asks the question: 'What number, when multiplied by itself, will give me the original number?' For example, when we consider the square root of 169, we use the symbol \( \sqrt{} \) and look for a number that yields 169 when squared. In this case, \( \sqrt{169} \) equals 13. It's important to note, however, that there's also a negative root to consider, as \( (-13)^2 \) also equals 169. This is because the act of squaring a negative number results in a positive product, just as with positive numbers.
When executing the square root operation, one must be aware that every positive number will indeed have two square roots: a positive and a negative one. This reflects the property that squaring either polarity of a number results in the same positive square.

Squaring Numbers

Squaring a number is a fundamental mathematical operation, represented by raising a number to the power of two. Notated as \( (number)^2 \), it means you multiply the number by itself. For instance, \( 5^2 = 5 \times 5 = 25 \). Squaring can apply to both positive and negative numbers, with both resulting in a positive product. The concept of squaring is pivotal when you reverse the operation to find the square root. This backward process identifies the original number before it was squared.

Example of Positive and Negative Squaring

Take the numbers 7 and -7. Squaring both, you get \( 7^2 = 49 \) and \( (-7)^2 = 49 \) as well. Despite the original numbers having different signs, the act of squaring them eliminates this difference, leaving identical results. This reinforces the necessity to check for both positive and negative roots when finding the square root of a positive number.

Mathematical Verification

The process of mathematical verification serves as a proof to confirm the correctness of solutions. Applied in the context of finding square roots, it means we take the proposed root and perform the operation that led to it—in this case, squaring the number—to see if it results in the original number we started with. If the verification is successful, the solution is concretely affirmed.
Returning to our example with the number 169, if we propose that 13 is one of its square roots, squaring 13 should give us 169. Similarly, for -13, squaring it must also produce 169. This verification \(13^2 = 169 \) and \( (-13)^2 = 169 \) substantiates that both 13 and -13 are valid square roots of 169. This step is not just a mechanical follow-through but a powerful illustration of understanding the relationship between an operation and its inverse, providing clear evidence of a mathematic truth.