Problem 43 - 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 \( 6s^3 \).

Step 1: Identify the Square Root

Determine that the square root expression is \( \sqrt{36s^6} \). The task is to simplify this under the assumption that the variables involved are unrestricted.

Step 2: Simplify the Numerical Part

Calculate the square root of 36, which is 6, because 6 multiplied by 6 equals 36. Thus the expression begins to simplify to \( 6 \sqrt{s^6} \).

Step 3: Simplify the Variable Part

Use the property \( \sqrt{s^{6}} = (s^{6})^{1/2} = s^{6/2} = s^{3} \). Since 6 divided by 2 is 3, this simplifies to \( s^3 \).

Step 4: Combine the Results

Combine the simplified components to get the final result of \( 6s^{3} \). Here, absolute value symbols are not necessary as \( s^3 \) is already simplified under the assumption that all variables are unrestricted.

Key Concepts

Understanding Square Roots

Square roots offer a way to reverse multiplication. They help us determine what number can be multiplied by itself to produce a given number.
For example, the square root of 36 is 6, because when you multiply 6 by itself, you get 36. This is often written as \( \sqrt{36} = 6 \). The symbol "\( \sqrt{} \)" is used to denote a square root.

When working with variables, square roots follow the same principle. If you have a variable raised to a power, such as \( s^6 \), the square root would involve dividing that exponent by 2. Thus, \( \sqrt{s^6} \) simplifies to \( s^{3} \).
This is because the exponent of 6 divided by 2 is 3.

1. For numerical parts like 36, find which number multiplied by itself gives 36.
2. For variable parts like \( s^6 \), divide the exponent by 2.

Square roots are powerful tools, especially in algebra, to simplify and manage expressions with both numbers and variables.

Algebraic Expressions: Simplifying with Variables

Algebraic expressions are combinations of numbers, variables, and operations. They can sometimes appear complex, but simplifying them can make these expressions manageable and useful for solving equations.

Consider an expression involving a number and a variable such as \( 6\sqrt{s^{6}} \). The goal in simplifying is to express this in the simplest form possible:

• First, tackle the constant or number part, like finding the square root of 36.
• Next, handle the variable by applying the appropriate mathematical operations, such as reducing exponents with square roots.

Finally, combine these simplified parts to form a clearer, streamlined expression. The result of simplifying \( \sqrt{36s^6} \) becomes \( 6s^{3} \).

Remember, when simplifying, consider the properties of exponents and roots to transform the expressions efficiently. The clearer your simplified expression, the easier it is to understand and use in further equations.

The Role of Exponents in Algebra

Exponents indicate how many times a number or variable is multiplied by itself. If you have \( s^6 \), it means \( s \) is multiplied by itself 6 times. Exponents are essential in algebra to manage large numbers with ease and to simplify expressions.

When simplifying expressions with exponents, such as \( s^6 \), especially under a square root, the rule is to divide the exponent by 2. This is based on the property \( (s^n)^{1/2} = s^{n/2} \). Hence, \( s^6 \) becomes \( s^{3} \) when under a square root.

• Exponents allow compact representation of large or repeated multiplications.
• Adjusting exponents according to root rules simplifies them for easier manipulation.

Being comfortable with exponents can significantly assist in tackling various algebraic problems efficiently, setting a foundation for more advanced mathematics.