Problem 81 - Elementary and 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

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

Step 1: Simplify the Fraction Inside the Square Root

Start by simplifying the expression inside the square root: \( \frac{125n^5}{64n} \). Divide the coefficients: \( \frac{125}{64} \) is already simplified. Now, divide the exponents on \( n \): \( n^{5-1} = n^4 \). The expression becomes \( \frac{125n^4}{64} \).

Step 2: Apply the Square Root to the Fraction

Apply the square root to both the numerator and the denominator separately: \( \sqrt{\frac{125n^4}{64}} = \frac{\sqrt{125n^4}}{\sqrt{64}} \). Recall \( \sqrt{64} = 8 \). Now the expression is \( \frac{\sqrt{125n^4}}{8} \).

Step 3: Simplify the Square Root of the Numerator

Simplify \( \sqrt{125n^4} \). Write 125 as a product of a perfect square: \( 125 = 25 \times 5 \). Therefore, \( \sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5} \). Next, \( \sqrt{n^4} = n^2 \). Thus, \( \sqrt{125n^4} = 5n^2\sqrt{5} \).

Step 4: Write Final Simplified Expression

Combine the simplified terms from the previous steps: place them over 8 to get the final expression \( \frac{5n^2\sqrt{5}}{8} \). This is the expression in its simplest form.

Key Concepts

Simplification

Simplification is an essential algebraic skill that involves reducing expressions to their simplest form. This process usually focuses on removing unnecessary terms or simplifying numerical fractions.

• Start by analyzing the expression or equation you have.
• In this exercise, we have a complex fraction inside a square root. The goal is to simplify it by reducing the powers and the coefficients.
• For the example provided, it began by simplifying the fraction \( \frac{125n^5}{64n} \). This is done by dividing both the coefficient \( 125 \) by \( 64 \) as much as possible, and simplifying the powers of \( n \) using the property \( n^a/n^b = n^{a-b} \).

To master simplification, practice applying these principles to various algebraic expressions.

Square Roots

Square roots are a special mathematical concept that involves finding a number which, when multiplied by itself, gives the original number.

• The process of dealing with square roots can often be simplified by focusing on the properties of perfect squares.
• In the provided exercise, we've applied the square root separately to the numerator and the denominator of the fraction.
• When dealing with the square root of a numerical fraction, such as \( \sqrt{64} \), recognize that 64 is a perfect square of 8, simplifying it directly to 8.

Similarly, breaking down complex numbers like 125 into \( 25 \times 5 \) helps in simplifying the square root of non-perfect squares. Always remember, simplifying square roots involves identifying and breaking down terms into their perfect square components whenever possible.

Exponents

Exponents are not only about powers; they are fundamental in simplifying algebraic expressions, particularly when they appear with variables.

• Understanding the rules of exponents is crucial. One basic rule is the subtraction of exponents in division: \( x^a/x^b = x^{a-b} \).
• In this exercise, you encountered the exponents when managing \( n^5/n \). With the subtraction rule, it reduces to \( n^{5-1} \) or \( n^4 \).
• Moreover, when taking square roots, existing exponents are essential. The square root of \( n^4 \) becomes \( n^{4/2} \), which simplifies to \( n^2 \).

Master these fundamental rules, and you'll handle complicated expressions with ease.