Problem 50 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\frac{8 y^{2 / 3} y^{-1}}{2^{-1} y^{3 / 4} y^{-1 / 6}}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{16}{y^{11/12}}\).

Step 1: Simplify the Numerator

Start by simplifying the numerator. We have \(8y^{2/3}y^{-1}\). Combine the exponents of \(y\) using the property that allows us to add exponents when multiplying like bases. Therefore, we have:\[y^{2/3 - 1} = y^{-1/3}\].The numerator simplifies to \(8y^{-1/3}\).

Step 2: Simplify the Denominator

Next, simplify the denominator \(2^{-1} y^{3/4} y^{-1/6}\). First, simplify the exponents of \(y\) using the property to add exponents:\[y^{3/4 - 1/6} = y^{9/12 - 2/12} = y^{7/12}\].Thus, the denominator becomes \(2^{-1}y^{7/12}\).

Step 3: Convert Negative Exponents

Convert the negative exponent \(2^{-1}\) to positive, which turns into \(\frac{1}{2}\). The denominator thus becomes \(\frac{1}{2}y^{7/12}\).

Step 4: Rewrite the Fraction

The expression is now:\[\frac{8y^{-1/3}}{\frac{1}{2}y^{7/12}}\] Dividing by a fraction is equivalent to multiplying by its reciprocal. This gives us:\[8y^{-1/3} \times 2y^{-7/12} = 16y^{-1/3}y^{-7/12}\]

Step 5: Simplify Exponents

Combine the exponents of \(y\) using the property that allows adding exponents:\[y^{-1/3 - 7/12}\]Convert \(-1/3\) to have a common denominator with \(7/12\):\[-1/3 = -4/12\]Thus, the exponents combine as:\[y^{-4/12 - 7/12} = y^{-11/12}\].

Step 6: Final Expression with Positive Exponents

The expression is now \(16y^{-11/12}\). Convert to using only positive exponents:\[\frac{16}{y^{11/12}}\]This is the final simplified expression.

Key Concepts

Simplifying Exponents

When simplifying expressions with exponents, it's important to remember the basic rules that govern how exponents work. These rules help us to combine and simplify terms effectively.
For instance, when you multiply terms with the same base, you simply add the exponents. In our example, the numerator simplifies by combining the exponents of the like base term, which is "y." Initially, we have terms like \(y^{2/3}\) and \(y^{-1}\). By adding these exponents, we get \(y^{2/3 - 1} = y^{-1/3}\).
On the other hand, when you encounter a negative exponent, it signifies the reciprocal of the base raised to the positive of that exponent. For instance, \(2^{-1}\) is equivalent to \(\frac{1}{2}\). Thus, simplifying exponents isn't just about numbers; the techniques apply uniformly regardless of the complexity of the expression.

Fraction Operations

Fraction operations involve rules that are crucial when manipulating algebraic fractions. In this problem, we start with a complex fraction.
To deal with it, first simplify any fractions in the numerator and denominator separately. After simplifying, manage the division between the fractions. Remember, dividing by a fraction is equivalent to multiplying by its reciprocal, which simplifies complex operations into multiplication tasks.
For example, in this exercise, the given fraction was \(\frac{8y^{-1/3}}{\frac{1}{2}y^{7/12}}\). We transform this by acknowledging that division is the same as multiplying by the reciprocal. Hence, \(\frac{8y^{-1/3}}{\frac{1}{2}y^{7/12}}\) becomes \(8y^{-1/3} \times 2y^{-7/12}\), making the calculations straightforward after this transformation.

Exponents and Powers

Understanding exponents and powers forms the foundation of many algebraic operations. Exponents tell us how many times a number, known as the base, is used as a factor in a multiplication.
In algebraic expressions, mastering how to convert and manage both negative and positive exponents is essential. While negative exponents might seem daunting, remember they simply represent the reciprocal of the number with a positive exponent.

• For example, \(y^{-x}\) becomes \(\frac{1}{y^x}\).
• Similarly, \(2^{-1}\) becomes \(\frac{1}{2}\).

During simplification, always aim to express final results using only positive exponents to meet standard conventions. In our problem, this meant converting from \(16y^{-11/12}\) to \(\frac{16}{y^{11/12}}\), ensuring the expression is in its simplest form with only positive exponents.