Problem 54 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{y^{11 / 3}}{\left(y^{5}\right)^{1 / 3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( y^2 \).

Step 1: Simplify the Denominator

The denominator is \( \left(y^{5}\right)^{1/3} \). To simplify, use the Power of a Power property \( (a^m)^n = a^{m \cdot n} \). Thus, \( \left(y^{5}\right)^{1/3} = y^{5 \cdot \frac{1}{3}} = y^{5/3} \).

Step 2: Simplify the Fraction

With the numerator \( y^{11/3} \) and the simplified denominator \( y^{5/3} \), apply the Quotient of Powers property \( \frac{a^m}{a^n} = a^{m-n} \). This gives us \( y^{11/3 - 5/3} = y^{(11/3 - 5/3)} = y^{6/3} \).

Step 3: Simplify the Exponent

The expression \( y^{6/3} \) can be further simplified. Divide 6 by 3 to get \( y^2 \). This is the expression simplified with positive exponents.

Key Concepts

Power of a Power Property

The "Power of a Power" property is one of the foundational properties of exponents.

It states that when you have an exponent raised to another exponent, like \((a^m)^n\), you can simplify it by multiplying the exponents together.

This property can drastically reduce the complexity of exponential expressions.In our example, we applied the "Power of a Power" property to simplify the denominator, \((y^5)^{1/3}\):

• Identify the base, which is \(y\).
• Recognize the first exponent, \(5\), and the second exponent, \(1/3\).
• Multiply these exponents to simplify: \(5 \times 1/3 = 5/3\).
• The expression \((y^5)^{1/3}\) becomes \(y^{5/3}\).

Using this property can simplify complex nested powers into a more manageable form. A common pitfall is to try adding exponents instead, so remember: multiplication is key here! Understanding this property will make simplifying exponential expressions more straightforward.

Quotient of Powers Property

The "Quotient of Powers" property is an essential rule that deals with dividing terms that have the same base.
When you divide powers with the same base, like \(\frac{a^m}{a^n}\), you subtract the exponents: \(a^{m-n}\).
This principle is valuable when you want to simplify fractions with exponential terms.In our given problem:

• The numerator is \(y^{11/3}\), and the denominator simplified to \(y^{5/3}\).
• Both terms have the same base \(y\), so we apply the property: \(y^{11/3 - 5/3}\).
• Subtract the exponents: \(11/3 - 5/3 = 6/3\).
• The result is \(y^{6/3}\).

The "Quotient of Powers" property makes dealing with fractions involving exponents simpler and helps monitor errors as calculation becomes straightforward. Remember, this is about subtracting exponents, not dividing or multiplying them.

Simplifying Expressions

Simplifying expressions is the process of making mathematical expressions as simple as possible.
This often means reducing them to a form that is easier to understand or work with.In the example given:

• We started with the expression \(\frac{y^{11/3}}{(y^5)^{1/3}}\).
• Using the "Power of a Power" property, the denominator became \(y^{5/3}\).
• With the "Quotient of Powers" property, the entire expression became \(y^{6/3}\).
• Finally, simplify the fraction \(6/3\) by dividing 6 by 3, which results in \(2\).
• Thus, \(y^{6/3}\) simplifies to \(y^2\).

Simplifying is crucial for solving equations, performing analyses, and understanding the nature of expressions.
It usually follows after applying various properties, like those of exponents. Always look for ways to simplify expressions to their most reduced form, which helps in making quick comparisons or calculations.