Evaluate (3^-2x^-5)/(y^0)

Solution:

Step 1: Simplify the given expression.

• Step 1.1: Apply the negative exponent rule, which states that $a^{-n} = \frac{1}{a^n}$, to convert $3^{-2}$ to the denominator. This results in $\frac{x^{-5}}{y^0 \cdot 3^2}$.

• Step 1.2: Similarly, apply the negative exponent rule to move $x^{-5}$ to the denominator, yielding $\frac{1}{y^0 \cdot 3^2 \cdot x^5}$.

Step 2: Simplify the denominator.

• Step 2.1: Recognize that any number raised to the power of zero is 1, simplifying to $\frac{1}{1 \cdot 3^2 \cdot x^5}$.

• Step 2.2: Calculate $3^2$ to get 9, simplifying further to $\frac{1}{1 \cdot 9 \cdot x^5}$.

• Step 2.3: Multiply 9 by 1 in the denominator to finalize the simplification, resulting in $\frac{1}{9 \cdot x^5}$.

Knowledge Notes:

The problem involves simplifying an algebraic expression using exponent rules. The relevant knowledge points for solving this problem include:

1. Negative Exponent Rule: For any nonzero number $b$ and integer $n$, $b^{-n} = \frac{1}{b^n}$. This rule is used to transform negative exponents into positive exponents by moving the base to the opposite side of the fraction.

2. Zero Exponent Rule: For any nonzero number $a$, $a^0 = 1$. This rule states that any number raised to the power of zero is equal to one.

3. Simplification of Fractions: When simplifying fractions, we combine the rules of exponents with basic arithmetic to reduce the expression to its simplest form.

4. Multiplication of Numbers: Multiplying any number by one does not change the value of the number. This is known as the multiplicative identity property.

By applying these rules systematically, the original expression is simplified step by step to its most reduced form.