Problem 41 - Intermediate Algebra

Question

Simplify each expression. Assume that all variables are unrestricted and use absolute value symbols when necessary. See Example 2. $$ \sqrt{81 h^{4}} $$

Step-by-Step Solution

Answer

\( \sqrt{81 h^4} = 9h^2 \).

Step 1: Identify the components of the expression

The expression given is \( \sqrt{81 h^4} \). We notice that this consists of a constant (81) and a variable raised to a power \( h^4 \).

Step 2: Simplify the constant under the square root

The square root of 81 can be simplified as \( \sqrt{81} = 9 \), since 81 is a perfect square (\( 9 \times 9 = 81 \)).

Step 3: Simplify the variable under the square root

The variable \( h^4 \) under the square root can be simplified as \( \sqrt{h^4} = h^2 \). This is because taking the square root of a variable raised to a power is equivalent to raising the variable to half of its original power: \( \frac{4}{2} = 2 \).

Step 4: Combine the results

Combine the simplified results of the constant and the variable: \( 9 \cdot h^2 = 9h^2 \). Since the variable \( h \) is unrestricted and \( h^2 \) is always positive, there's no need for absolute value symbols.

Key Concepts

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 81 is 9 because 9 multiplied by 9 equals 81. Symbolically, this is represented as \( \sqrt{81} = 9 \). Notice how this process simplifies more complex expressions.

When dealing with square roots, especially in algebra, simplifying

• Determine if the number is a perfect square.
• Use the property of exponents if variables are involved.

Using these steps ensures you maintain a clear and accurate simplification. Perfect squares within the radical are easy to simplify. This makes handling complex expressions much more straightforward.

Algebraic Expressions Defined

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication). They form the building blocks of algebra, allowing you to represent real-world situations and solve problems numerically. Consider the expression \( 81h^4 \). It embodies a constant (81) and a variable expression \( h^4 \). Such expressions allow complex mathematical manipulations.

In simplifying expressions:

• Identify constants and variables separately.
• Understand the operation connecting them, in this case, multiplication.
• Simplify step-by-step, handling each part systematically.

Remember, simplifying algebraic expressions often involves reducing their complexity, making them easier to work with. This involves breaking them down into known components, such as perfect squares and simple variable terms.

Demystifying Variables and Exponents

Variables represent unknown or changeable values, usually denoted by letters like \( h \). They allow us to create general formulas applicable in many situations. In the expression \( h^4 \), the variable \( h \) is raised to an exponent, 4. This indicates \( h \times h \times h \times h \).

Exponents denote repeated multiplication of a number by itself. Understanding exponents helps tremendously in simplifying expressions under a square root.

• Rewrite the term with the exponent: \( h^4 = (h^2)^2 \).
• Apply the square root: \( \sqrt{h^4} = h^2 \).

This process reveals how exponents reduce from multiplication to addition when taking the root: halving the exponent gives the power of 2. This interplay simplifies even the most daunting-looking expressions.