Problem 64 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{y^{-3}}{y^{4}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{y^7} \).

Step 1: Understanding the Expression

The expression given is \( \frac{y^{-3}}{y^{4}} \). This is a fraction where the base \( y \) has exponents in both the numerator and denominator.

Step 2: Applying the Quotient Rule for Exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Apply this to simplify \( \frac{y^{-3}}{y^{4}} \) to \( y^{-3-4} = y^{-7} \).

Step 3: Converting Negative Exponent to Positive Exponent

To remove the negative exponent, recall that \( a^{-m} = \frac{1}{a^m} \). Therefore, \( y^{-7} \) becomes \( \frac{1}{y^7} \).

Key Concepts

Quotient Rule for Exponents

When you encounter exponents within a fraction, you might find yourself puzzled, but don't worry! The quotient rule for exponents is here to make life easier. This rule is a nifty tool that helps simplify expressions like \( \frac{y^{-3}}{y^{4}} \). In simple terms, the quotient rule tells us how to divide powers with the same base.

Here's how it works:

• When you divide two exponents with the same base, \( a^m \) and \( a^n \), you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

Applying this rule to \( \frac{y^{-3}}{y^{4}} \), you simply subtract the exponent in the denominator from the one in the numerator, resulting in:

• \( y^{-3-4} = y^{-7} \)

This reduces our equation to an expression with a single exponent— much simpler, right?

Negative Exponents

Negative exponents can seem strange at first, but they're a breeze once you get the hang of them. If you see a negative exponent, such as \( y^{-7} \), it indicates a reciprocal.

Here's the basic idea:

• A negative exponent \( a^{-m} \) is equal to \( \frac{1}{a^m} \). It flips the base to the other side of a fraction.

So when we simplify \( y^{-7} \), we convert it to a positive exponent:

• \( y^{-7} = \frac{1}{y^{7}} \)

This transformation helps remove negative exponents from equations, making solutions more conventional and easier to handle. Never leave a negative exponent in your final answer  instead, convert it to a positive exponent by taking the reciprocal.

Positive Exponents

Positive exponents, unlike their negative cousins, are straightforward. They show how many times a base is multiplied by itself.

If you have an expression like \( y^7 \), it simply means:

• The base \( y \) is used as a factor seven times: \( y \times y \times y \times y \times y \times y \times y \).

Positive exponents are preferred in final answers because they typically represent the number in its simplest, most understandable form.

Whenever you simplify an expression with exponents, aim to shift any negative exponents to positive ones by moving them to the denominator using the reciprocal rule. This step ensures your answer is clear and in line with standard mathematical conventions.