Problem 123 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{300 m^{5}}{64}} $$

Step-by-Step Solution

Answer

\(\frac{5m^2 \text{√3m}}{4}\)

Step 1: Break Down the Fraction

Separate the fraction under the square root into the numerator and the denominator: \[\frac{\text{numerator}}{\text{denominator}} = \frac{300 m^5}{64}\]

Step 2: Simplify the Numerator

Factorize the numerator to identify perfect squares: \[300 m^5 = 3^1 \times 100 m^4 \times m = 3^1 \times (10^2)^2 \times m^4 \times m\]Here, \(100 = 10^2\), and thus \(m^5 = m^4 \times m\).

Step 3: Simplify the Denominator

Rewrite the denominator as a perfect square:\[64 = 8^2\]

Step 4: Apply the Square Root

Simplify inside the square root by applying the square root to both the numerator and the denominator separately: \[\text{Simplify} \frac{\text{√(300 m^5)}}{\text{√64}} = \frac{\text{√(3^1 \times (10^2)^2 \times m^4 \times m)}}{\text{8}}\]

Step 5: Simplify the Square Roots

Take the square roots of the factors individually: \[\text{√(300 m^5)} = \text{√(3^1)} \times \text{√((10^2)^2)} \times \text{√(m^4)} \times \text{√(m)} = \text{√3} \times 10 \times m^2 \times \text{√m}\]and\[\text{√64} = 8\]

Step 6: Combine the Results

Combine all simplified terms:\[\frac{\text{√3} \times 10 \times m^2 \times \text{√m}}{8} = \frac{10m^2 \text{√3} \text{√m}}{8} = \frac{10m^2 \text{√3m}}{8}\] Finally, simplify the fraction:\[\frac{10m^2 \text{√3m}}{8} = \frac{5m^2 \text{√3m}}{4}\]

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of rewriting an algebraic expression in a simpler form. This often involves combining like terms, factoring expressions, and reducing fractions. In this example, we simplify the expression \(\frac{300m^5}{64}\) under a square root.
Let's break it down step by step to make it easier to understand:

• Separate the numerator and the denominator.
• Identify and factorize perfect squares.
• Apply the square root to both parts, simplifying each term individually.

This approach can help turn complex expressions into ones that are much easier to work with.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\text{√64} = 8\) because \(\text{8 × 8 = 64}\).
When simplifying square roots, especially within fractions, here’s what to do:

• Identify and isolate perfect squares within the numerator and denominator.
• Use the property \(\text{√(a/b)} = \frac{\text{√a}}{\text{√b}}\).

For the given exercise, the denominator is already a perfect square: \(\text{64 = 8}^2\). For the numerator, factorize to find perfect square factors and simplify accordingly. For example, \(\text{300m}^5 \rightarrow 3 \times 100 \times m^4 \times m \). By simplifying, the square roots of each term can be individually taken for further reduction.

Fraction Simplification

Fraction simplification involves reducing fractions to their lowest terms. This process becomes significant when dealing with square roots.
Follow these steps to simplify fractions under a square root:

• Separate the square root across the numerator and denominator.
• Factorize both the numerator and the denominator.
• Take the square roots of individual factors.
• Simplify the resulting terms and reduce the final fraction, if possible.

Using our example, start by simplifying \(\frac{\text{√(300 m^5)}}{\text{√64}} \rightarrow \frac{\text{√3} × 10 × m^2 × \text{√m}}{8}\). Then reduce the result to its simplest form, \(\frac{5m^2 \text{√3m}}{4}\).

Factoring

Factoring is the process of breaking down a number into its prime factors or a polynomial into its polynomial factors.
In our example, factoring helps simplify the expression inside the square root.

• Identify factors that are perfect squares.
• Rewrite the expression to show these factors.
• Once the expression is factored, simplify by taking the square root of the perfect squares.

For example, factorizing \(\text{300m}^5 = 3 \times 100 \times m^4 \times m\), we can simplify individual parts: \(\text{√3} \times 10 \times m^2 \times \text{√m}\). This makes it much more manageable to simplify further.