Problem 105 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ -\sqrt{49 b^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-7b^4\).

Step 1: Identify the core components

The expression we need to simplify is \(-\sqrt{49b^8}\). This expression contains a square root applied to the term \(49b^8\), with an additional negative sign in front of the root.

Step 2: Simplify the numerical part

Recognize that 49 is a perfect square, as \(49 = 7^2\). So, the square root of 49 is 7.

Step 3: Simplify the variable part

The expression inside the square root is \(b^8\). Applying the square root to this expression: \(\sqrt{b^8} = b^{8/2} = b^4\).

Step 4: Combine the simplified components

Now combine the simplifications: the square root of \(49b^8\) becomes \(7b^4\). Don't forget the negative sign in front of the original expression, so we attach it to the result to have \(-7b^4\).

Key Concepts

Square Roots

The square root is a vital concept in mathematics, particularly when simplifying radical expressions. Essentially, finding the square root of a number means determining which number multiplied by itself will result in the original number. For example, since \(7 \times 7 = 49\), the square root of 49 is 7. When dealing with square roots, you'll often encounter numbers that are not perfect squares, meaning they do not have an integer as a square root. In such cases, the square root will either remain in its radical form or can sometimes be approximated to a decimal.
Square roots are denoted by the radical symbol \(\sqrt{}\), with the number or expression placed underneath. It's crucial to apply the square root to both numerical values and variables separately. By isolating these components, as we did with \(\sqrt{49b^8}\), you can simplify each part correctly and then combine them for the final simplified expression.

Perfect Squares

Understanding perfect squares is essential for simplifying expressions involving square roots. A perfect square is an integer that results from an integer multiplied by itself. For instance, 1, 4, 9, 16, and 49 are all perfect squares because they equal \(1^2\), \(2^2\), \(3^2\), \(4^2\), and \(7^2\) respectively. Recognizing these numbers can make computations more straightforward, especially when simplifying expressions.

• Perfect squares allow you to swiftly simplify the numerical component of a radical expression. Recognizing that 49 is \(7^2\), we know \(\sqrt{49} = 7\).
• Using perfect squares diminishes the complexity of the original radical, leading to simplified and manageable results.

In mathematics, especially algebra, knowing and identifying perfect squares saves time and effort during problem-solving.

Variable Exponents

Variable exponents often appear in algebraic expressions, and understanding them is key to simplifying square roots involving these variables. An exponent indicates how many times a number, or in this case, a variable, is multiplied by itself. For example, \(b^8\) means that the variable \(b\) is multiplied by itself eight times.
When you encounter exponents under an even-root radical such as a square root, the strategy involves dividing the exponent by the root's index, which is 2 for square roots. Therefore, \(\sqrt{b^8}\) becomes \(b^{8/2} = b^4\).

• This division process is due to a rule of exponents: \((b^m)^n = b^{m\times n}\).
• It is important to perform this division carefully to arrive at the correct simplification.

Whenever simplifying expressions with variable exponents, always consider whether a square root or any even-root operation can help reduce the expression to its simplest form.