Problem 48 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{\sqrt{75 y^{3}}}{\sqrt{3 y}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 5y \).

Step 1: Simplify the Numerator

The numerator of the expression is \( \sqrt{75y^3} \). Start by factoring 75 into its prime factors: \( 75 = 3 \times 5^2 \). Thus,\[ \sqrt{75 y^3} = \sqrt{3 \times 5^2 \times y^3}. \]Identify the perfect squares under the square root: \( 5^2 \) and \( y^2 \). Simplify it to:\[ \sqrt{3} \times 5 \times y^{3/2} = 5y^{3/2} \sqrt{3}. \]

Step 2: Simplify the Denominator

The denominator is \( \sqrt{3y} \), which can be rewritten as:\[ \sqrt{3} \times \sqrt{y} = \sqrt{3y} = 3^{1/2} y^{1/2}. \]

Step 3: Divide the Expressions

Now, divide the simplified numerator by the simplified denominator:\[ \frac{5y^{3/2} \sqrt{3}}{(3y)^{1/2}} = \frac{5y^{3/2} \sqrt{3}}{3^{1/2} y^{1/2}}. \]

Step 4: Apply Cancellation

Cancel the common terms between numerator and denominator:1. The \( \sqrt{3} \) cancels out between numerator and denominator.2. Simplify the remaining \( y \) terms:\[ \frac{5y^{3/2}}{y^{1/2}} = 5y^{(3/2) - (1/2)} = 5y^1. \]

Step 5: Final Simplification

Thus, after cancellations and simplification, the expression simplifies to:\[ 5y. \]

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. Understanding these is crucial when dealing with expressions under radical signs, as seen in the exercise. To simplify radical expressions effectively:

• Identify any perfect squares or cubes within the expression as these can be easily simplified.
• The goal is to express the expression in its simplest form.
• In the given exercise, we simplified \( \sqrt{75y^3} \) by recognizing that 75 can be broken down into \( 3 \times 5^2 \). This allowed us to take \( 5^2 \) out of the square root, simplifying our work.

Recognizing these steps and practicing with different radical expressions can greatly enhance understanding.

Prime Factorization

Prime factorization is the process of breaking down a number into its smallest prime factors. In algebra, this concept is utilized when simplifying expressions and working with radicals.

• Understanding prime numbers—numbers greater than 1 that have no divisors other than 1 and themselves—is essential.
• Begin with the smallest prime number and see if it divides the number, then move to the next if it doesn't. Repeat until you break down the entire number.
• For example, the number 75 in the exercise was factored into \( 3 \times 5^2 \). Identifying these factors allowed us to determine the component parts of the original expression, aiding in its simplification.

This technique is incredibly useful for breaking down complex expressions and working with radical forms.

Exponent Rules

Exponent rules help us manage the powers of numbers or expressions efficiently. These rules simplify operations involving exponents, making it easier to handle expressions like those in our exercise.

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

In the provided exercise, when faced with the division of expressions, we used the rule by subtracting exponents: \( \frac{y^{3/2}}{y^{1/2}} \) led us to \( y^{1} \). This understanding of exponent rules allows us to manipulate and simplify complex algebraic expressions effectively. Familiarizing oneself with these rules is key to mastering algebraic simplification.