Problem 36 - Elementary Algebra

Question

For the following problems, find the two square roots of the given number. $$ 10,000 $$

Step-by-Step Solution

Answer

Answer: 100 and -100

Step 1: Identifying the given number

The given number is 10,000.

Step 2: Applying the square root property

To find the two square roots of 10,000, we apply the square root property: ±√10,000

Step 3: Calculating the square roots

Using a calculator or by observation, find the positive square root of 10,000: $$ \sqrt{10,000} = 100 $$ Now we can determine the negative square root by taking the negative of the positive square root: $$ - \sqrt{10,000} = -100 $$

Step 4: Writing the final answer

The two square roots of 10,000 are: 100 and -100.

Key Concepts

Positive and Negative Roots

When dealing with square roots, it is important to understand that every positive number has two square roots: one positive and one negative. This might seem a bit puzzling at first, but here's an easy way to grasp it.

The square root function essentially "undoes" squaring a number. So, when we think of square roots, we are looking for a number that, when multiplied by itself, gives the original number. For example, both 10 and -10 give 100 when squared ( \(10 \times 10 = 100 \) and \(-10 \times -10 = 100 \) ). This is why the number 100 has two square roots: 10 and -10.

Remember that while the principal square root refers to the positive square root, in many cases, both positive and negative roots are significant, especially in solving equations. Hence, for 10,000, the square roots are ±100, representing both positive and negative roots.

• Positive square root: refers to the non-negative value that squares back to the original number.
• Negative square root: is simply the negative value of the positive square root.

Square Root Calculation

Calculating square roots may seem daunting at first, but it can be straightforward when using the right methods and tools. For easy-to-square numbers like 10,000, observing the number pattern can help:

### Example of Calculation
The square root of a number like 10,000 can be calculated directly due to its round and recognizable nature. It ends with a certain number of zeros (four in this case), which is significant. A number ending with an even number of zeros, such as 100, 1,000, or 10,000, is a perfect square.
It is often useful to break it down: \( 10,000 = 100^2 \), hence the square root is 100. You can use a calculator for immediate results or estimate manually by identifying simpler patterns for large benchmarks.

• The process is deducing the root by identifying pairs of tens.
• Understand it as a product of smaller multiplication groups.

Moreover, a systematic way is through attempting factorization methods if observations don't suffice. However, for numbers like 10,000, recognizing the zero pattern is easiest and most efficient.

Mathematical Properties

Understanding the mathematical properties related to square roots can simplify many mathematical problems. Square roots follow certain rules and proprieties, such as:

• **Product Rule:** \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). You can multiply the square roots of two numbers and get the same as the square root of their product.
• **Quotient Rule:** \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) for \(b eq 0\). You can take the square root of a fraction by dividing the square roots of the numerator and the denominator.
• **Zero Property:** The square root of zero is zero (\( \sqrt{0} = 0 \)), and there's only one square root for zero as the positive and negative vice doesn't apply here.

Additionally, knowing that square roots return non-negative values as principal roots (except when specifically noting the negative) assists in managing equations or inequalities effectively. These properties enhance the flexibility and depth of knowledge necessary for using square roots in broader mathematics contexts.