Problem 61 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{y^{4}}{y^{5}}\)

Step-by-Step Solution

Answer

\( \frac{1}{y} \)

Step 1: Identify the problem

We need to simplify the expression \( \frac{y^4}{y^5} \) by using the properties of exponents. The goal is to express this without any negative exponents.

Step 2: Apply the Property of Exponents

Recall the rule for dividing powers with the same base: \( \frac{a^m}{a^n} = a^{m-n} \). This means we should subtract the exponent of the denominator from the exponent of the numerator.

Step 3: Calculate the Exponent

Subtract the exponent in the denominator (5) from the exponent in the numerator (4): \( y^{4-5} = y^{-1} \).

Step 4: Simplify with Positive Exponents

Rewrite \( y^{-1} \) using positive exponents. According to the exponent rules, \( a^{-n} = \frac{1}{a^n} \). Thus, \( y^{-1} = \frac{1}{y^1} \) or simply \( \frac{1}{y} \).

Key Concepts

Properties of Exponents

Exponents are powerful tools in algebra that simplify complex calculations. Understanding the properties of exponents is crucial because they provide clear rules for manipulating expressions involving powers. Here are the key properties:

• Product of Powers Property: When multiplying two powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Property: When dividing two powers with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Property: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m\times n} \).
• Zero Exponent Rule: Any base with an exponent of zero equals one: \( a^0 = 1 \), where \( a eq 0 \).
• Negative Exponent Rule: A negative exponent represents a reciprocal: \( a^{-n} = \frac{1}{a^n} \).

These rules help us to simplify expressions in a systematic way and are particularly helpful when dealing with algebraic equations.

Dividing Powers

When faced with dividing powers, especially in an expression like \( \frac{y^4}{y^5} \), the properties of exponents come into play. To simplify, we rely on the quotient of powers property.

The **Quotient of Powers Property** tells us that we should subtract the exponent of the denominator from that of the numerator when the bases are the same. It's a straightforward process, simplifying what could otherwise be a cumbersome task. Applying this property to our example:

• Identify the base, which is \( y \) in this case.
• Subtract the exponents: \( 4 - 5 = -1 \).

This results in \( y^{-1} \). The calculation is straightforward but requires attention to ensure exponents are correctly handled whenever dividing similar bases.

Positive Exponents

Simplifying expressions to ensure that all exponents are positive is a common requirement in algebra. Positive exponents make calculations and interpretations easier, as they directly represent how many times a number is multiplied by itself.When dealing with negative exponents, such as when we simplify \( y^{-1} \), we employ the concept of reciprocals, which is guided by the negative exponent rule:

• Negative exponents denote an inverse. For example, \( y^{-1} \) becomes \( \frac{1}{y^1} \).
• It is often simplified further to just \( \frac{1}{y} \), removing the implied power of one.

Keeping all exponents positive is not just a format requirement but also enhances clarity and avoids potential errors in later steps of solving algebra problems. This approach ensures the expressions are cleaner and align with mathematical conventions.