In Exercises 91–100, simplify using properties of exponents. (3y^1/4)³/y^1/12

Verified step by step guidance

1. Step 1: Start by applying the power rule of exponents to the numerator. The power rule states that \((a^m)^n = a^{m \cdot n}\). Here, \((3y^{1/4})^3\) becomes \(3^3 \cdot y^{(1/4) \cdot 3}\).
2. Step 2: Simplify the terms in the numerator. \(3^3\) simplifies to \(27\), and \(y^{(1/4) \cdot 3}\) simplifies to \(y^{3/4}\). So the numerator becomes \(27y^{3/4}\).
3. Step 3: Rewrite the expression as \(\frac{27y^{3/4}}{y^{1/12}}\). Now, use the quotient rule of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\), to simplify the \(y\)-terms.
4. Step 4: Subtract the exponents of \(y\) in the numerator and denominator. \(y^{3/4 - 1/12}\). To subtract these fractions, find a common denominator. The least common denominator (LCD) of 4 and 12 is 12. Rewrite \(3/4\) as \(9/12\), so the subtraction becomes \(9/12 - 1/12 = 8/12\).
5. Step 5: Simplify the fraction \(8/12\) to \(2/3\). The final simplified expression is \(27y^{2/3}\).

Key Concepts

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these properties is essential for simplifying expressions with exponents.

Simplifying Radicals

Simplifying radicals involves rewriting expressions with roots in a simpler form. For example, y^(1/4) can be expressed as the fourth root of y. This concept is important when dealing with fractional exponents, as it allows for easier manipulation and simplification of expressions involving roots and powers.

Fractional Exponents

Fractional exponents represent both powers and roots. For instance, y^(1/4) indicates the fourth root of y, while y^(3/4) represents y raised to the third power and then taking the fourth root. Recognizing how to interpret and manipulate fractional exponents is crucial for simplifying complex expressions involving them.