Problem 60 - Algebra 1: Concepts and Skills

Question

Simplify the expression. The simplified expression should have no negative exponents. (Lesson 8.4). $$ \left(\frac{y^{5}}{y^{7}}\right)^{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \(y^{4}\).

Step 1: Apply the Quotient Rule

The quotient rule states we should subtract the exponents when dividing same bases. Therefore \(y^{5}/y^{7}= y^{5-7} = y^{-2}\).

Step 2: Raise to the Power -2

Now, raise \(y^{-2}\) to the exponent -2. According to the power of a power rule, we should multiply the exponents. So \((y^{-2})^{-2}= y^{-2*(-2)} = y^{4}\).

Step 3: Simplify Result

The result of simplification is a single term: \(y^{4}\).

Key Concepts

Quotient Rule

The quotient rule is a basic principle in algebra that helps you simplify expressions where you are dividing terms with the same base. In simpler terms, when you have two expressions with the same base that are being divided, you can subtract the exponent of the denominator from the exponent of the numerator.
This rule can be expressed as:

• \(\frac{a^m}{a^n}=a^{m-n}\)

Here, \(a\) is the common base, and \(m\) and \(n\) are the respective exponents.
For example, given \(\frac{y^{5}}{y^{7}}\), you subtract the exponents \(5 - 7\), resulting in \(y^{-2}\).
This greatly simplifies the expression and is extremely useful when handling polynomials and rational functions.

Negative Exponents

Negative exponents might seem daunting, but they're just another way to express division. Essentially, a negative exponent indicates that you should take the reciprocal of the base raised to the positive of that exponent.
For instance, \(a^{-n}\) means \(\frac{1}{a^n}\).
When you see an expression such as \(y^{-2}\), you should think of it as \(\frac{1}{y^2}\).

• Negative exponents allow us to express everything in one line, making it easier to understand and work with complex algebraic expressions.
• In the example given, the negative exponent signals that the term \(y^2\) would be in the denominator if not raised to another power subsequently.

Remember, converting negative exponents to positive is always your target when simplifying, as it gives a clearer view of the real value of an expression.

Power of a Power Rule

The power of a power rule is used when you have an exponent raised to another exponent. This rule states that you multiply the exponents to simplify the expression. The general form is:

• \((a^m)^n = a^{m \times n}\)

For example, if you have \((y^{-2})^{-2}\), you multiply \(-2\) by \(-2\) to get \(4\), resulting in \(y^4\).
This rule helps tidy up expressions, especially when dealing with nested exponents or complex algebraic terms. Following the power of a power rule consistently leads to cleaner and more comprehensible expressions.
In the original problem, applying this rule helped convert a negative exponent into a positive one, simplifying the expression to \(y^4\). Using this technique is crucial for ensuring your work is precise and enhances your ability to work with higher powers confidently.