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Chapter 10: Problem 225

In the following exercises, simplify. $$\frac{y^{4}}{y^{20}}$$

Short Answer

Expert verified
The simplified form is \(\frac{1}{y^{16}}\).

Step by step solution

01

Understand the Problem

The task requires simplifying the expression \(\frac{y^{4}}{y^{20}}\). This means we need to reduce the given fraction to its simplest form using the properties of exponents.
02

Apply the Quotient of Powers Rule

Use the quotient of powers rule, which states \(\frac{a^m}{a^n} = a^{m-n}\), to simplify \(\frac{y^{4}}{y^{20}}\). So, \(\frac{y^{4}}{y^{20}} = y^{4-20}\).
03

Simplify the Exponent

Compute the exponent by subtracting: \(4 - 20 = -16\). Thus, the expression becomes \(y^{-16}\).
04

Rewrite with Positive Exponent

To express \(y^{-16}\) with a positive exponent, use the negative exponent rule which states \(a^{-m} = \frac{1}{a^m}\). Hence, \(y^{-16} = \frac{1}{y^{16}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quotient of powers rule
In algebra, simplifying expressions with exponents can be made easier with the quotient of powers rule. This rule applies when you divide two exponents with the same base. It states that
  • \(\frac{a^m}{a^n} = a^{m-n}\)
      • This means you subtract the exponent in the denominator from the exponent in the numerator.
      For example, if you have
      • \(\frac{y^4}{y^{20}}\)
          • according to the quotient of powers rule:
          • \(\frac{y^4}{y^{20}} = y^{4-20} = y^{-16}\)
              • By subtracting 20 from 4, you get -16, which simplifies the fraction to \(y^{-16}\). The quotient of powers rule is a great shortcut for handling such operations.
negative exponent rule
Dealing with negative exponents may seem confusing at first, but the concept is straightforward. The negative exponent rule helps to convert negative exponents into positive ones. The rule states that
  • \(a^{-m} = \frac{1}{a^m}\)
      • This means that any number with a negative exponent can be written as its reciprocal with a positive exponent.
      For example, from our simplified expression in the earlier section:
        \(y^{-16}\)
          This can be rewritten using the negative exponent rule:
          • \(y^{-16} = \frac{1}{y^{16}}\)
              • Thus, instead of dealing with negative exponents, you flip the base to the denominator, making it easier to handle.
              The negative exponent rule is crucial for simplifying expressions in a comprehensible and standardized manner.
fraction simplification
Simplifying fractions in algebra involves reducing expressions to their simplest form. It's important to work through each step systematically for clarity.
When starting with our initial problem:
  • \(\frac{y^4}{y^{20}}\)
      • We applied the quotient of powers rule to simplify it into \(y^{-16}\).
      To further simplify it into a more recognizable form, we used the negative exponent rule:
      • \(y^{-16} = \frac{1}{y^{16}}\)
          • Thus, our final simplified fraction is \(\frac{1}{y^{16}}\).
          Simplifying fractions with exponentiation may initially seem complex; however, by applying the correct rules step by step, it becomes more approachable and easier to manage. Utilizing these fundamental techniques allows anyone to simplify and understand these algebraic expressions effectively.

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