Problem 45 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ -\sqrt{b^{12}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression \(-\sqrt{b^{12}}\) is \(-b^6\).

Step 1: Observe the given expression

We are given the expression: $$ -\sqrt{b^{12}} $$ Our task is to simplify this expression.

Step 2: Simplify the square root

The expression inside the square root is \(b^{12}\). Since 12 is an even number, we can simplify the square root as follows: $$ \sqrt{b^{12}} = b^{12/2} = b^6 $$

Step 3: Include the negative sign

Now, we need to include the negative sign that was outside the square root: $$ -\sqrt{b^{12}} = -b^6 $$

Step 4: Final answer

The simplified expression is: $$ -\sqrt{b^{12}} = -b^6 $$

Key Concepts

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. In mathematics, the square root of a number \( x \) is usually denoted by \( \sqrt{x} \). It is essential to know that the square root and squaring are opposite operations. This means if you apply both operations in succession on a number, you'd return to the original number, i.e., \( \sqrt{x^2} = x \).

• Finding Square Roots: For perfect squares like 4, 9, 16, etc., the square roots are whole numbers. For non-perfect squares, the square roots are often irrational numbers.
• Operations involving Square Roots: When dealing with expressions inside a square root, aim to simplify as much as possible. This involves breaking down numbers or expressions into their smallest factors or base components.

Understanding ##### Example ComparisonIn the problem we're looking at, \( \sqrt{b^{12}} \), the task was to simplify the expression inside the square root first. Since the exponent 12 is a multiple of 2, it allowed for an exact division, simplifying the inside expression to \( b^6 \). After this, we considered the negative sign as shown above. This will always result in a clear, simplified output.

Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \( b^{12} \), the base is \( b \) and it is raised to the 12th power.

• Property of Exponents: One key property is \( (b^m)^n = b^{m\cdot n} \). This means when you raise a power to another power, you multiply the exponents.
• Application: When taking the square root of an exponent, you divide the exponent by 2, such as \( \sqrt{b^{12}} = b^{12/2} = b^6 \).

Exponents can greatly simplify algebraic expressions by reducing the number of terms. In our example, noticing that 12 divides perfectly by 2 allows us to turn \( b^{12} \) into \( b^6 \) without changing the base, hence simplifying the expression neatly.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. This involves removing parentheses, combining like terms, and reducing complicated expressions while retaining the original value.

• Simplifying Radical Expressions: When simplifying radical expressions, identify perfect square factors, use exponent rules, and then reduce any coefficients appropriately.
• Dealing with Negative Signs: A negative sign outside an expression such as \( -\sqrt{x} \) means that after simplification, maintain the negative multiplier.

In the original exercise, simplification required understanding the exponent and square root relationship, reducing \( \sqrt{b^{12}} \) to \( b^6 \) and including the provided negative sign to achieve the finalized result \( -b^6 \). Thus, a clear understanding of algebraic rules helps maintain the integrity and simplicity of the expression.