Problem 99 - College Algebra Essentials

Question

In Exercises \(91-100,\) simplify using properties of exponents. $$\frac{\left(3 y^{\frac{1}{4}}\right)^{3}}{y^{\frac{1}{12}}}$$

Step-by-Step Solution

Answer

The answer is \(27 y^{\frac{2}{3}}\).

Step 1: Simplify The Numerator

The first step will be to simplify the exponent inside the numerator. The given expression is \(\frac{\left(3 y^{\frac{1}{4}}\right)^{3}}{y^{\frac{1}{12}}}\). The exponent inside the numerator can be simplified by applying the power rule. The power rule states that when raising a power to a power, you multiply the exponents. So, \(3^{3} = 27\) and \((y^{\frac{1}{4}})^{3} = y^{\frac{3}{4}}\). So, the numerator becomes \(27 y^{\frac{3}{4}}\). The expression now is \(\frac{27 y^{\frac{3}{4}}}{y^{\frac{1}{12}}}\).

Step 2: Apply the Division Rule For Exponents

The next action to take is to simplify the fraction by applying the division rule for exponents. This rule states that when you divide terms with the same base, you subtract the exponents. In this case, we subtract \(\frac{1}{12}\) from \(\frac{3}{4}\). To be able to perform this subtraction, remember that \(\frac{3}{4}\) can be rewritten as \(\frac{9}{12}\). So \(\frac{9}{12} - \frac{1}{12} = \frac{8}{12} = \(\frac{2}{3}\). Thus, \( y^{\frac{2}{3}}\). So the expression simplifies to \(27 y^{\frac{2}{3}}\).

Step 3: Final Simplification

At this point, there is no additional simplification possible, so the function has been effectively simplified.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics. It involves reducing expressions to their simplest form. This often makes equations easier to work with. In algebra, simplification can include combining like terms and reducing fractions. For this exercise, you'll see how properties like the power rule and division rule for exponents help streamline the process. By applying these rules systematically, we transform the original expression into a simpler one that's easier to interpret.
Understanding how to simplify expressions is crucial in not only solving problems more efficiently, but also in recognizing patterns that can help you solve more complex equations later.

Power Rule

The power rule in exponents is a fundamental concept. It states that when you raise a power to another power, you multiply the exponents. This rule is very handy when expressions involve multiple layers of powers. For example:

• Given \((3 y^{\frac{1}{4}})^3\), apply the rule to simplify the expression to \((3^3) (y^{\frac{3}{4}})\).
• This results in \(27 y^{\frac{3}{4}}\).

The power rule helps to break down complex expressions into more manageable parts. This makes it easier to process and further reduce expressions.

Division Rule for Exponents

The division rule for exponents is a key tool in simplifying expressions with exponents. This rule tells us that when dividing terms with the same base, we subtract the exponents.
Consider the expression:

• From \(\frac{27 y^{\frac{3}{4}}}{y^{\frac{1}{12}}}\), apply the division rule by subtracting the exponents: \(\frac{3}{4} - \frac{1}{12}\).
• Convert \(\frac{3}{4}\) to \(\frac{9}{12}\) to make the subtraction easier.
• Perform the subtraction to get \(y^{\frac{8}{12}} = y^{\frac{2}{3}}\).

This method significantly reduces the complexity of the expression, making it simpler to manage.

Fraction Subtraction

Subtracting fractions is a skill that often comes in handy when working with exponents. To perform subtraction, you need a common denominator. Here's a quick guide:

• Convert the fractions to have the same denominator. For example, convert \(\frac{3}{4}\) to \(\frac{9}{12}\) so that it can be subtracted from \(\frac{1}{12}\).
• Subtract the numerators: \(9 - 1 = 8\).
• The result is \(\frac{8}{12}\), which simplifies to \(\frac{2}{3}\).

This process is essential when simplifying expressions with exponents and ensures clarity and accuracy in calculations.