Question

Prep Exercise 9 Every positive integer has two square roots, a ____ square root and a ____ square root. Prep Exercise 10 A negative number has ____ real square roots. find all square roots of each number. $$49$$

Short answer

#tag_title# Prep Exercise 11 Answer #tag_content# The two square roots of 49 are 7 (positive square root) and -7 (negative square root).

Step-by-step solution

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9.

Types of Square Roots

Every positive integer has two square roots: a positive square root and a negative square root. Negative numbers, on the other hand, do not have any real square roots, because when a negative number is multiplied by itself, it becomes a positive number. Now, let's answer the main questions.

Finding the Square Roots of 49

To find the two square roots of 49, think of two numbers that, when multiplied by themselves, equal 49. The positive square root is found as follows: $$\sqrt{49} = 7$$ This is because $$7 \times 7 = 49$$. For the negative square root, we have: $$-\sqrt{49} = -7$$ This is because $$-7 \times -7 = 49$$.

Prep Exercise 9 Answer

Every positive integer has two square roots: a positive and a negative square root.

Prep Exercise 10 Answer

A negative number has no real square roots.

Key concepts

Positive Square Root

When we discuss the square root of a number, we are essentially looking for a number that produces the original number when it is multiplied by itself. The positive square root is the non-negative value that fulfills this condition. For example, when we calculate the square root of 36, we get 6, because 6 times 6 equals 36.

It's crucial to recognize that for any positive number, there's always a positive square root. These positive roots are often represented with the radical symbol without a negative sign. In most real-world contexts, when people refer to the square root, they are often thinking of the positive square root by default.

Negative Square Root

Less frequently discussed but just as valid, is the negative square root. Every positive number has a negative counterpart that, when squared, will yield the original number. To denote the negative square root, we precede the radical symbol with a negative sign.

For instance, the negative square root of 64 is -8 because \(-8 \times -8 = 64\). Despite their lesser use in everyday scenarios, negative square roots play a significant role in various mathematical contexts, such as solving polynomial equations or in the field of complex numbers.

Real Square Roots

Talking about real square roots introduces us to the concept of real numbers, which include all the numbers that we typically use: positive numbers, negative numbers, and zero. Real square roots are the square roots that result in real numbers. As such, all positive numbers have two real square roots - one positive and one negative as we have seen.

However, it's essential to highlight that negative numbers do not have real square roots because squaring a real number always results in a non-negative product. Therefore, any operation to square root a negative number takes us out of the realm of real numbers and into the world of complex numbers.

Finding Square Roots

When you're tasked with finding square roots, there are several methods you could use. A straightforward method is to consider what number multiplied by itself equals the number for which we're finding the square root. This is often simplest with perfect squares like 1, 4, 9, 16, and so on. In other cases, we can use square root algorithms or a calculator for an approximate value. For example, let’s look at the number 49. The square root of 49 is 7 because \(7 \times 7 = 49\).

To find square roots without a calculator, you can use long division style algorithms or estimation. It's important to keep in mind that accuracy improves with these methods as you carry out more steps. Whether you are dealing with perfect squares or not, understanding the underlying concepts is crucial in finding both the positive and negative roots.