Problem 22 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{\frac{36}{z^{4}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{6}{z^2} \).

Step 1: Simplify the fraction under the square root

The fraction under the square root is \( \frac{36}{z^4} \). We can simplify this by taking the square root of the numerator and the square root of the denominator separately. Start with the numerator, where \( \sqrt{36} = 6 \).

Step 2: Simplify the denominator

Now, apply the square root to the denominator. The square root of \( z^4 \) is \( z^2 \) because \( (z^2)^2 = z^4 \). So, \( \sqrt{z^4} = z^2 \).

Step 3: Combine the results

Combine the results from Step 1 and Step 2 to get the simplified expression. This gives us \( \frac{6}{z^2} \) after simplifying \( \sqrt{\frac{36}{z^4}} \) as \( \frac{\sqrt{36}}{\sqrt{z^4}} = \frac{6}{z^2} \).

Key Concepts

Square Roots

Square roots can initially seem mysterious, but they're just asking, "What number, when multiplied by itself, gives me this number?" When you see a square root symbol, it's seeking the number that fits this description. For example, the square root of 36, written as \( \sqrt{36} \), is 6. Why? Because multiplying 6 by itself (i.e., 6 * 6) results in 36. Breaking down square roots into their components can simplify the process substantially.

• For perfect squares like 36 or 49, the square root will result in an integer.
• Not all numbers have perfect integer square roots — numbers like 2 or 3 will have square roots that are not whole numbers.
• To handle square roots in a fraction, divide each part separately, as \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Understanding how to break down numbers under the square root symbol is key to mastery.

Fraction Simplification

Fractions can look complicated, but simplifying them makes math operations clearer and easier. When simplifying a fraction, you are basically making the numbers as small as possible without changing the overall value. This involves finding any common factors in the numerator and the denominator.

• Start by identifying numbers that can divide both the top (numerator) and bottom (denominator).
• Dividing the numerator and the denominator by these common factors "cancels" them out.
• A simplified fraction is not always "small" in numerical value, but it's less cluttered and easier to use in calculations.

In the exercise, we approached it by examining the square root of a fraction separately, which was \( \sqrt{\frac{36}{z^4}} = \frac{\sqrt{36}}{\sqrt{z^4}} \). By doing this, it becomes simple to simplify each part step by step.

Exponent Rules

Understanding exponent rules helps immensely when simplifying algebraic expressions that contain powers. Exponentiation is simply a shortcut to represent repeated multiplication. For instance, \( z^4 \) means \( z \times z \times z \times z \).
A few core rules will make handling exponents straightforward:

• When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing like bases, subtract the exponents: \( a^m / a^n = a^{m-n} \).
• Taking an exponent to another power multiplies them: \((a^m)^n = a^{m \cdot n} \).

In this problem, we used the simple rule for a square root of an exponent: \( \sqrt{z^4} \). Here, you're essentially doing \((z^2)^2 = z^4 \), meaning the square (\( 2 \)) cancels out the exponent (also \( 2 \)), resulting in \( z^2 \). This makes simplifying such expressions more manageable.