Problem 93 - College Algebra

Question

In Exercises \(85-94,\) simplify using properties of exponents. $$\frac{\left(3 y^{2 / 4}\right)^{3}}{y^{1 / 12}}$$

Step-by-Step Solution

Answer

The final simplified form of the expression is \(27y^{17/12}\)

Step 1: Simplify the Expression in the Numerator

The property of a power of a power states that \( (a^m)^n = a^{m \times n} \). Using this property, the complex power in the numerator can be simplified. Here, the expression \((3 y^{2 / 4})^3\) is simplified as such: \(3^3 \times (y^{2/4})^3 = 27y^{(2/4)\times 3} = 27y^{3/2}\)

Step 2: Invoke Division Property of Exponents

The division property of exponents says that when you divide numbers with the same base, you subtract the exponent in the denominator from the exponent in the numerator: \( a^{m}/a^{n} = a^{m-n}\). We apply this property in this exercise. So, \(27y^{3/2} / y^{1 / 12} = 27y^{3/2 - 1/12}\)

Step 3: Transform Fraction Exponents to Common Basis

Before subtracting the two exponents in the previous step, we have to transform these fractions to equivalent fractions with common denominators. The exponents \(3/2\) and \(1/12\) then become \(18/12\) and \(1/12\) respectively. Now we have \(27y^{18/12 - 1/12} \)

Step 4: Subtract the Exponents

Lastly, subtract the exponents from each other to obtain \(27y^{17/12}\)

Key Concepts

Power of a Power Property

When working with powers raised to another power, things can be simplified by merging the exponents. This rule is known as the power of a power property. In mathematical terms, it looks like this: \((a^m)^n = a^{m \times n}\). For instance, if you have \((3y^{2/4})^3\), you can simplify by applying this property:

• First, deal with the constant: \(3^3 = 27\).
• Then handle the variable: \((y^{2/4})^3 = y^{(2/4) \times 3} = y^{3/2}\).

This makes the expression much easier to deal with in subsequent steps.

Division Property of Exponents

The division property of exponents helps simplify expressions where you divide numbers with the same base. If the bases are the same, you simply subtract the exponents: \(a^m / a^n = a^{m-n}\).
Using this with our example \(27y^{3/2} / y^{1/12}\), focus on the exponents:

• Subtract the denominator’s exponent from the numerator’s: \(3/2 - 1/12\).

You'll simplify it further by getting a common denominator for those fractions.

Fractional Exponents

Fractional exponents might look tricky, but they're just another way to express roots. Here’s how they work:

• The fraction’s numerator is the power.
• The fraction’s denominator is the root.

Consider \(y^{3/2}\) and \(y^{1/12}\). To simplify, convert them to have a common denominator. For \(3/2\) and \(1/12\), the least common denominator is 12, rewriting them as \(18/12\) and \(1/12\). This makes subtraction easy in the next step.

Simplification of Expressions

Once you've aligned everything with common denominators, you're all set to simplify the expression. Let’s follow through with our example:

• After rewriting the exponents, we had \(y^{18/12}\) and \(y^{1/12}\).
• Subtract them: \(18/12 - 1/12 = 17/12\).

This results in \(27y^{17/12}\). Always aim to break down complex expressions into simpler parts, making calculations straightforward!