Introduction to Radical Expressions: Videos & Practice Problems

Understanding square roots is essential for solving exponential expressions where the base is unknown. The square root of a positive number has two values: a positive principal root and a negative root, represented by the radical symbol $\sqrt{x}$ for the positive root and a minus sign before the radical for the negative root. Negative numbers inside radicals indicate imaginary numbers, as no real number squared yields a negative result. This concept is crucial when working with polynomials, exponents, and terms in standard form, ensuring accurate interpretation of roots and coefficients in algebraic expressions.

Square Roots

In mathematics, understanding the relationship between exponents and square roots is crucial. When we square a number, such as 4, we find that \(4^2 = 16\). Conversely, if we are given the result, like 16, and asked to find the original number that, when multiplied by itself, yields 16, we utilize square roots. The square root operation is essentially the inverse of squaring a number.

For example, to determine the square roots of 9, we seek a number that, when squared, equals 9. Testing various integers, we find that both 3 and -3 satisfy this condition since \(3^2 = 9\) and \((-3)^2 = 9\). Thus, 9 has two square roots: 3 (the principal root) and -3. This principle holds true for all positive real numbers, which will always have two square roots.

The notation for square roots is represented by the radical symbol (√). When you see √9, it refers specifically to the positive root, which is 3. To denote the negative root, we write -√9, indicating -3. It is important to note that writing ±√9 is incorrect in this context, as it implies both roots simultaneously, which can lead to confusion in mathematical expressions.

To illustrate further, consider evaluating √36. We need to find a number that, when squared, equals 36. Testing integers, we find that \(6^2 = 36\), so √36 equals 6. Since the radical is presented without a negative sign, we only consider the positive root. Conversely, if we see -√36, this indicates the negative root, which is -6.

Now, when dealing with negative numbers inside a radical, such as √(-36), we encounter a different scenario. No real number squared will yield a negative result, meaning that the square root of a negative number is not defined within the realm of real numbers. Instead, it is classified as an imaginary number. This distinction is essential: a negative outside the radical (e.g., -√9) is valid, while a negative inside (e.g., √(-9)) indicates an imaginary result.

In summary, understanding square roots involves recognizing that they are the inverse of squaring a number, identifying both positive and negative roots, and knowing how to properly use notation to avoid confusion. Additionally, recognizing the implications of negative numbers within radicals is crucial for accurate mathematical interpretation.

Evaluate the radical.
$\sqrt{{(-5)}^2}$

• A. $2.23$
• B. $5$
• C. $-5$
• D. No real solution (Imaginary)

Answer: (B)

Evaluate the following square root.
$\sqrt{144}$

• A. $10$
• B. $14$
• C. $12$
• D. $16$

Answer: (C)

Evaluate the following square root.
$-\sqrt{121}$

• A. $11i$
• B. $11$
• C. $-11$
• D. $-10$

Answer: (C)

Determine if the given square root evaluates to a real number.
$-\sqrt{4}$

• A. Real
• B. Not Real
• C. Cannot be determined

Answer: (A)

Determine if the given square root evaluates to a real number.
$\sqrt{-64}$

• A. Real
• B. Not Real
• C. Cannot be determined

Answer: (B)

Determine if the given square root evaluates to a real number.
$-\sqrt{-25}$

• A. Real
• B. Not Real
• C. Cannot be determined

Answer: (B)

Here’s what students ask on this topic:

What is the definition of a square root in algebra?

In algebra, a square root of a number is a value that, when multiplied by itself, equals the original number. For example, the square root of 9 is a number that satisfies the equation $\sqrt{9}=\; x$ such that $x^2=9$. There are always two square roots for positive real numbers: a positive root called the principal root and a negative root. For 9, these roots are 3 and -3 because both $3^2=\; 9$ and ${-}^3=\; 9$. The radical symbol $\sqrt{}$ denotes the principal (positive) root, while a negative sign before the radical indicates the negative root. Understanding this concept is fundamental for solving equations involving exponents and radicals.

How do you interpret the radical symbol when solving square root problems?

The radical symbol $\sqrt{}$ represents the principal (positive) square root of a number. For example, $\sqrt{16}=4$ means the positive root 4, not -4. If you want to indicate the negative root, you place a negative sign before the radical, like $-\sqrt{16}=-4$. Sometimes, to represent both roots simultaneously, the notation $\pm \sqrt{16}$ is used, meaning both +4 and -4. It is important not to write $\sqrt{16}=\; \pm 4$ because the radical symbol alone implies only the positive root. Proper notation helps avoid confusion and errors when solving algebraic problems involving square roots.

Why can't you take the square root of a negative number in real numbers?

In the real number system, you cannot take the square root of a negative number because no real number squared results in a negative value. For example, $3^2=\; 9$ and ${-}^3=\; 9$, both positive. Squaring any real number always produces a non-negative result. Therefore, expressions like $\sqrt{-9}$ are undefined in real numbers and are called imaginary or complex numbers. This concept is important because it signals when you need to extend your number system beyond real numbers, which is covered in more advanced algebra courses. For now, if you see a negative inside the radical, it means the expression is imaginary.

How do you evaluate square roots of perfect squares like 36?

To evaluate the square root of a perfect square such as 36, you find the number that, when multiplied by itself, equals 36. Since $6^2=\; 36$, the principal square root is 6. The radical expression $\sqrt{36}$ represents this positive root, so $\sqrt{36}=6$. If you want the negative root, you write $-\sqrt{36}=-6$. Remember, the radical symbol alone always refers to the positive root. This method applies to all perfect squares, making it straightforward to evaluate their square roots.

What is the difference between a negative sign inside and outside the radical symbol?

The placement of a negative sign inside versus outside the radical symbol changes the meaning of the expression. A negative sign outside the radical, such as $-\sqrt{9}$, means the negative of the principal square root, which equals -3. This is a valid real number. However, a negative sign inside the radical, like $\sqrt{-9}$, asks for the square root of a negative number, which is not defined in the real numbers and is considered imaginary. This distinction is crucial because it affects whether the result is a real or imaginary number. Remember: negative outside the radical is okay; negative inside the radical means the expression is imaginary.