Problem 32 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt{100 c^{4}}, c \geq 0 $$

Step-by-Step Solution

Answer

The expression simplifies to \( 10c^2 \).

Step 1: Rewrite the Expression

First, simplify the expression by breaking down the square root of the product. The expression \( \sqrt{100 c^4} \) can be rewritten as \( \sqrt{100} \times \sqrt{c^4} \).

Step 2: Evaluate the Square Roots

Calculate the square root of each part separately. The square root of 100 is 10, and the square root of \( c^4 \) is \( c^2 \) (since \((c^2)^2 = c^4\)). Therefore, the expression simplifies to \( 10c^2 \).

Step 3: Consider the Constraint

The constraint \( c \geq 0 \) is essential because it ensures that \( c^2 \) is non-negative, preserving the validity of the square root operation over real numbers.

Step 4: Combine the Results

The simplified expression is \( 10c^2 \), considering all given constraints and simplifications have been satisfied.

Key Concepts

Real Numbers

When evaluating expressions, like \(\sqrt{100 c^{4}}\), it is crucial to understand the concept of real numbers. Real numbers include all the numbers on the continuous number line like positive and negative integers, fractions, and decimals. They can be further divided into rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot.
Real numbers play a significant role in simplifying expressions, especially when dealing with square roots. The square root function is defined over non-negative real numbers. Therefore, to keep calculations valid, the expression must remain within the set of real numbers. This ensures that when we simplify or evaluate expressions, we are performing operations that make sense in the context of real numbers.

Square Root Simplification

Simplifying square roots is an essential skill in algebra. It involves breaking down the square root of a number or expression into simpler components. In the expression \(\sqrt{100 c^{4}}\), we use the property that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to rewrite the expression as \(\sqrt{100} \times \sqrt{c^{4}}\).
This allows us to handle each part separately:

• The square root of 100 is 10, because 10 multiplied by itself gives 100.
• The square root of \(c^{4}\) is \(c^{2}\), as \((c^{2})^{2} = c^{4}\).

By breaking it down this way, we transform a more complex expression into a simpler one: \(10c^{2}\). Simplification not only makes it easier to understand but also to work with further in mathematical calculations.

Non-negative Constraints

When evaluating expressions, constraints ensure that the operations are valid and meaningful. In this particular exercise, the constraint \(c \geq 0\) is introduced to ensure non-negativity.
Let's delve into why this is crucial:

• Real numbers under the square root must be non-negative to yield a real number result. This is because the square root of a negative number is not real, instead, it results in an imaginary number.
• This constraint ensures the expression \(c^{2}\) remains non-negative, as squaring any non-negative number always gives a non-negative result.

By adhering to the constraint \(c \geq 0\), we confirm that all operations stay in the realm of real numbers, preserving the integrity of solutions and making it possible to confidently use the simplified final result of \(10c^{2}\).