Problem 47 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{y^{1 / 3}}{y^{1 / 6}} $$

Step-by-Step Solution

Answer

The simplified expression is \( y^{\frac{1}{6}} \).

Step 1: Recall the properties of exponents

When you have a quotient of two expressions with the same base, you can subtract the exponent in the denominator from the exponent in the numerator: \( \frac{a^m}{a^n} = a^{m-n} \).

Step 2: Subtract the exponents

Using the property from Step 1, subtract the exponent \( \frac{1}{6} \) from \( \frac{1}{3} \): \[ \frac{1}{3} - \frac{1}{6} = \frac{2}{6} - \frac{1}{6} = \frac{1}{6} \].

Step 3: Write the simplified expression

The expression simplifies to \( y^{\frac{1}{6}} \). This is the expression with the positive exponent.

Key Concepts

Quotient Rule of Exponents

When dealing with exponents in a fraction, the quotient rule of exponents helps us simplify the expression. This rule states that when you divide two expressions with the same base, you subtract the exponents: if you have \( \frac{a^m}{a^n} \), it becomes \( a^{m-n} \).
This rule simplifies the process by reducing the need for multiplying out numbers, making your work easier and faster.

• Ensure both top and bottom of the fraction share the same base.
• Subtract the exponent of the denominator from the exponent of the numerator.
• The result will be the base raised to the new exponent.

This method is not only efficient but also reduces potential errors when working with larger equations or when variables are involved.

Negative and Positive Exponents

Exponents can be either positive or negative, and knowing how to interpret them is key in simplifying expressions. A positive exponent indicates how many times you multiply the base by itself. For example, \( a^3 \) means \( a \times a \times a \). A negative exponent, such as \( a^{-2} \), signals that you take the reciprocal of the base and then apply the exponent: \( a^{-2} = \frac{1}{a^2} \).

• Positive exponents represent standard multiplication.
• Negative exponents indicate reciprocal use and inversion of multiplication.
• A zero exponent means the base equals 1 (e.g., \( a^0 = 1 \)).

Remember, when the problem asks for positive exponents, it's important to convert any negative exponents by taking the reciprocal, ensuring the final expression only involves non-negative exponents.

Simplifying Expressions with Exponents

To simplify expressions with exponents, you align the application of exponent rules carefully. Start with identifying the common bases and applying appropriate exponent laws, like the quotient rule, product rule, and power rule. By setting up the expression correctly and breaking it down step-by-step, you can reduce even complex expressions into simpler forms.
Here's a simple guideline:

• Identify common bases in the expression.
• Use the exponent rules—quotient, product, and power—to combine and simplify exponents.
• Ensure all exponents are positive by addressing negatives as reciprocals when necessary.

This systematic approach helps in not only simplifying quickly but also avoids mistakes that might come from missing a key exponent rule or misapplying them. Remember, practice with different expressions will make these steps feel like second nature.