Problem 43 - Elementary and Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{36 s^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \(6|s^3|\).

Step 1: Identify the Parts of the Expression

Start by recognizing what is inside the square root. We have \(36s^6\), where 36 is a constant and \(s^6\) is a variable term raised to the sixth power.

Step 2: Simplify the Constant

Take the square root of 36. Since the square root of 36 is 6, simplify this part of the expression as \(6\).

Step 3: Simplify the Variable Term

Next, take the square root of \(s^6\). Remember that when you take the square root of a power, you divide the exponent by 2. So, \(\sqrt{s^6} = s^{6/2} = s^3\).

Step 4: Combine Simplified Parts

Combine the simplified parts from Steps 2 and 3 together. This yields \(6s^3\).

Step 5: Consider Absolute Values

Normally, for even powers under a square root, absolute value must be considered if variables can take any real value. Since \(s^6\) originally comes from \(s^3\) twice, ensure that the simplified version makes \(s^3\) always positive: \(|s^3|\).

Key Concepts

Variables with Exponents

When handling algebraic expressions, particularly those involving square roots, it's common to encounter variables with exponents. A variable with an exponent, such as \( s^6 \), indicates that the variable \( s \) is to be multiplied by itself a specific number of times. Here, \( s^6 \) means \( s \times s \times s \times s \times s \times s \).
To simplify expressions with variables raised to powers when taking the square root, divide the exponent by two. For our example, \( \sqrt{s^6} \) becomes \( s^{6/2} = s^3 \). This process reduces the power while maintaining the relationship contained in the original variable expression.

Absolute Value in Algebra

In algebra, the concept of absolute value is crucial when simplifying expressions involving even powers and roots. The absolute value of a number refers to its distance from zero on a number line, without considering direction, meaning it is always non-negative.
When simplifying square roots of variables with even exponents, for example \( s^6 \), the simplified result should always be non-negative. This leads to expressions like \( |s^3| \), ensuring the value is always positive. By using absolute values, we account for the possibility of \( s \) being either positive or negative, maintaining the solution's validity for all real numbers.

Power and Root Operations

Power and root operations often come together in mathematics. When simplifying square roots involving exponents, understanding how to work with both is essential. Taking the square root of an expression involves finding a number that, when multiplied by itself, gives the original number.
For instance, the square root of 36 is 6, because \( 6 \times 6 = 36 \). For variables, if you have \( \sqrt{s^6} \), this involves dividing the power by two, resulting in \( s^3 \).
Remember, whenever you simplify expressions involving roots, always consider absolute values when necessary. This maintains the integrity of the original expression, especially when variables are involved that could represent both positive and negative numbers.