Problem 81 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{\frac{125 n^{5}}{64 n}} $$

Step-by-Step Solution

Answer

\( \frac{5 n^2 \sqrt{5}}{8} \) is the simplified expression.

Step 1: Simplify Inside the Radical

Simplify the fraction inside the square root. Divide the terms in the numerator by the terms in the denominator: \[ \frac{125 n^{5}}{64 n} = \frac{125}{64} \cdot \frac{n^5}{n} = \frac{125}{64} \cdot n^{4}\]Thus, the expression becomes: \[ \sqrt{\frac{125}{64} \cdot n^{4}}\]

Step 2: Simplify the Fraction

Separate the square root of the fraction into the square roots of the numerator and the denominator: \[ \sqrt{\frac{125}{64} \cdot n^4} = \sqrt{\frac{125}{64}} \cdot \sqrt{n^4}\]The expression becomes: \[ \frac{\sqrt{125}}{\sqrt{64}} \cdot \sqrt{n^4}\]

Step 3: Evaluate the Square Roots of Constants

Find the square root of both numbers inside the fraction since they are perfect squares: \[ \sqrt{64} = 8, \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}.\]So: \[ \frac{\sqrt{125}}{\sqrt{64}} = \frac{5\sqrt{5}}{8}.\]

Step 4: Simplify the Variable Expression

Find the square root of the variable expression: \[ \sqrt{n^4} = n^2.\]Thus the expression becomes: \[ \frac{5\sqrt{5}}{8} \cdot n^2\]

Step 5: Combine the Final Expression

Multiply the results from the constants and the variable: \[ \frac{5n^2\sqrt{5}}{8}.\]This is the simplified form of the original expression.

Key Concepts

Fraction Simplification

Simplifying fractions is a pivotal skill in working with more complex algebraic expressions. It involves reducing a fraction to its simplest form. This means you divide the numerator and the denominator by their greatest common factor. In the case of our exercise, we first simplified the fraction \( \frac{125n^5}{64n} \). Each term is divided separately:

• The numerical part: \( \frac{125}{64} \) remains as it is until further breakdown.
• The variable part: \( \frac{n^5}{n} = n^4 \) by applying the rule \( a^m/a^n = a^{m-n} \).

By simplifying, we reduced the fractional part to \( \frac{125}{64} \cdot n^4 \), easing the calculation within the radical.

Radical Expressions

Radical expressions often involve square roots. These can initially look complex, but by breaking them down into simpler parts, they're much easier to handle. When simplifying \( \sqrt{\frac{125}{64} \cdot n^4} \), we separated components into individual square roots:

• \( \sqrt{\frac{125}{64}} \) which becomes an individual challenge to simplify.
• \( \sqrt{n^4} \) which quickly resolves to \( n^2 \) because \( \sqrt{a^2} = a \).

The reduction of radicals depends on finding perfect square factors, as in the constant factor of 125, rewritten as \( \sqrt{25 \times 5} = 5\sqrt{5} \). Thus, the radical simplifies systematically into an easier-to-use expression.

Exponent Rules

Working with exponents is essential in simplifying expressions like \( n^5/n \). Exponent rules allow us to combine and simplify terms efficiently.

• Here's a key rule: \( a^m/a^n = a^{m-n} \). This is used to reduce \( n^5/n \) to \( n^4 \).
• Another important rule is \( (a^m)^n = a^{mn} \) when dealing with higher order powers.

Exponent rules also play a role when dealing with radicals since \( \sqrt{n^4} \) simplifies to \( n^2 \). Remember that some roots can be rewritten as fractional exponents, like \( \sqrt{a} \) being \( a^{1/2} \), which broadens the application of exponent rules across different forms of algebraic expressions.