Problem 107 - Algebra and Trigonometry

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$ \frac{\left(x^{-2} y\right)^{-3}}{\left(x^{2} y^{-1}\right)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Rewrite expression using the power of a power rule

The power of a power rule states that \((a^m)^n = a^{m*n}\), we should apply this rule first. The expression becomes \( x^{(-2*-3)} * y^{(-3)} \div x^{(2*3)} * y^{(-1*3)}\).

Step 2: Simplify expressions

We simplify the exponent multiplication in the next step. It becomes \( x^{6} * y^{-3} \div x^{6} * y^{-3}\).

Step 3: Apply exponent rules for multiplication and division

For terms with the same base, we subtract the exponents if it is division and add if it is multiplication. The expression becomes \(x^{6-6} * y^{-3-(-3)}\).

Step 4: Calculate the new exponentials

Calculate the exponents. The expression simplifies to \(x^{0} * y^{0}\). Since any number (except zero) raised to the power of zero is 1, both \(x^{0}\) and \(y^{0}\) equal to 1.

Step 5: Multiply the results

Finally, we multiply 1 by 1, which is 1. The final simplified expression is 1.

Key Concepts

Power of a Power Rule

The "power of a power rule" is a handy tool when dealing with exponents. This rule helps you simplify expressions that involve an exponent raised to another exponent. It states that \((a^m)^n = a^{m \times n}\). This means you multiply the exponents together.

For example, if you have \((x^{-2})^{-3}\), apply the rule by multiplying the exponents: \(-2 \times -3 = 6\). This simplification transforms the expression into \(x^6\).

This rule is a shortcut to avoid writing more extended expressions and is especially useful in complex equations.

Exponent Rules

Exponent rules govern how we handle mathematical expressions involving exponents.

Some essential exponent rules include:

• Multiplication Rule: When multiplying terms with the same base, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Division Rule: When dividing terms with the same base, subtract the exponents: \(a^m \div a^n = a^{m-n}\).
• Zero Exponent Rule: Any number (except zero) raised to the power of zero equals 1: \(a^0 = 1\).

These rules are vital in simplifying expressions, especially when dealing with fractions or multiple terms with exponents.

Simplifying Expressions

Simplifying expressions involves using exponent rules to reduce a complex expression to its simplest form.

Start by applying the power of a power rule to resolve any nested exponents. Next, simplify the terms individually using addition or subtraction of exponents when multiplying or dividing terms with the same base.

For example, given an expression \(x^{6} \cdot y^{-3} \div x^{6} \cdot y^{-3}\), simplify it by subtracting exponents for division: \(x^{6-6}\) and \(y^{-3 - (-3)}\). This results in \(x^0 \cdot y^0\). Each term equals 1, so the entire expression simplifies to 1.

Simplifying expressions transforms complicated problems into manageable ones, making calculations more straightforward.