Problem 57 - College Algebra with Modeling and Visualization

Question

Exercises \(55-64:\) Use the power rules to simplify the expression. Use positive exponents to write your answer. $$ \left(y^{4}\right)^{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{y^8} \).

Step 1: Apply the Power of a Power Rule

The expression \( \left(y^{4}\right)^{-2} \) involves applying a power to a power. According to the power of a power rule \( (a^m)^n = a^{m \times n} \), we multiply the exponents. Multiplying 4 by -2, we get \( y^{4 \times (-2)} = y^{-8} \).

Step 2: Convert to Positive Exponents

The expression \( y^{-8} \) contains a negative exponent. To convert it to a positive exponent, we use the rule \( a^{-m} = \frac{1}{a^m} \). Thus, \( y^{-8} = \frac{1}{y^8} \).

Key Concepts

Exponents

Exponents are an elegant way of expressing repeated multiplication of the same number. When you see a number like \( y^4 \), it means \( y \) is multiplied by itself 4 times, or \( y \times y \times y \times y \). Exponents can be integers, fractions, and even negative numbers, each affecting the base number (until now \( y \) in our case) in its own special way. The larger the exponent, the more times the base multiplies itself. Exponents are crucial for simplifying expressionsbecause they allow compact and comprehensive representation of potentially extensive multiplication.

Power of a Power Rule

The Power of a Power Rule is a shortcut that simplifies expressions where an exponent is raised to another exponent.The rule is straightforward:

• In an expression like \((a^m)^n\), you multiply the exponents: \(m\) and \(n\).
• So, \((a^m)^n = a^{m \times n}\).

In our example, \((y^4)^{-2}\), we apply this rule by multiplying 4 and -2, resulting in \(y^{-8}\).This rule efficiently reduces multiple layers of exponents into a single, more manageable expression. Understanding this will help simplify expressions whenever you face nested exponents.

Negative Exponents

Negative exponents introduce an interesting twist: they imply division instead of multiplication. For any number with a negative exponent, say \( a^{-m} \), you reframe it as the reciprocal of the number with a positive exponent:

• \( a^{-m} = \frac{1}{a^m} \)

This rule is crucial when converting expressions to avoid negative exponents. For instance, \(y^{-8}\) translates to \(\frac{1}{y^8}\). This understanding helps in maintaining the expression with positive exponents, which is generally preferred in simple mathematical solutions.

Simplifying Expressions

Simplifying expressions is about making them more concise and aesthetically pleasing. It also makes future calculations easier. By applying algebraic rules like the Power of a Power Rule and the negative exponent conversion, you make expressions like \((y^4)^{-2}\) more manageable. To simplify with positive exponents:

• Apply the Power of a Power Rule to combine exponents.
• Convert any negative exponents into positive by taking the reciprocal.

The final result \(\frac{1}{y^8}\) demonstrates a simple, clear representation of the original expression.Following these steps also ensures that the expression is ready for further operations or evaluations. Simplification is key to mathematical efficiency and clarity.