Problem 55 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \left(x y^{5} z^{-2}\right)^{-1} $$

Step-by-Step Solution

Answer

\(\frac{z^2}{xy^5}\)

Step 1: Apply the Negative Exponent Rule

The expression given is \((xy^5z^{-2})^{-1}\). According to the negative exponent rule, \((a^m)^{-n} = a^{-mn}\). Thus, we apply this rule by inverting the base and changing the sign of the exponent, resulting in \((xy^5z^{-2})^{-1} = x^{-1}y^{-5}z^{2}\).

Step 2: Simplify the Expression

Now simplify the expression \(x^{-1}y^{-5}z^{2}\). To write it with only positive exponents, move each term with a negative exponent to the denominator. The simplified expression is: \(\frac{z^2}{xy^5}\).

Key Concepts

Negative Exponent Rule

The negative exponent rule is a crucial concept in mathematics that deals with exponents. Understanding this rule can make simplifying expressions much easier. This rule states that if you have a base raised to a negative exponent, such as \( a^{-n} \), you can rewrite it as \( \frac{1}{a^n} \). In essence, a negative exponent indicates that the base should be moved to the opposite part of the fraction (from numerator to denominator or vice versa) and the exponent becomes positive in the process. This is why

• \( 2^{-3} = \frac{1}{2^3} \)
• \( (xy^5z^{-2})^{-1} = x^{-1}y^{-5}z^2 \)

By correctly applying the negative exponent rule, we can change expressions to a more manageable form with positive exponents, which is often required for further mathematical operations.

Simplifying Expressions

Simplifying expressions means making them easier to understand and work with. This often involves reducing the expression to its most basic form. In the context of exponents, this process often includes converting negative exponents to positive ones and combining similar terms.

In the given exercise, we started with \( (xy^5z^{-2})^{-1} \) and applied the negative exponent rule to rewrite it as \( x^{-1}y^{-5}z^2 \). The next step in simplifying is to ensure all exponents are positive by moving terms to the correct part of the fraction.

• Terms with negative exponents in the numerator move to the denominator and become positive.
• For example, \( x^{-1} \) becomes \( \frac{1}{x} \) and \( y^{-5} \) becomes \( \frac{1}{y^5} \).

By following these rules, the expression simplifies to \( \frac{z^2}{xy^5} \), where all exponents are positive.

Positive Exponents

Working with positive exponents is much simpler than dealing with negative ones. Positive exponents indicate how many times you multiply the base by itself. This makes calculations and simplifications more straightforward. In simplified expressions, having positive exponents helps avoid confusion and lays the groundwork for further mathematical operations.

For example:

• With an expression like \( z^2 \), you simply multiply \( z \) by itself, which is clear and intuitive.
• Thus, \( \frac{z^2}{xy^5} \) is easily understood as \( z \times z \) over \( x \) times \( y \) multiplied by itself five times.

Keeping exponents positive is a standard practice, and converting negative to positive exponents simplifies working with and communicating mathematical ideas, ensuring clarity in problem-solving.