Simplify $\frac{x^{5} y^{-2}}{(x^3 y)^2}$.
Understand the Problem
The question requires simplifying an algebraic expression involving exponents. We need to apply the rules of exponents to simplify the expression and determine the correct option from the given choices.
Answer
A. $x^5y$
Answer for screen readers
A. $x^5y$
Steps to Solve
- Apply the power of a product rule to the numerator
The power of a product rule states that $(ab)^n = a^n b^n$. Applying this to the numerator $(x^2y)^3$, we get: $$ (x^2y)^3 = (x^2)^3 \cdot y^3 $$
- Apply the power of a power rule
The power of a power rule states that $(a^m)^n = a^{mn}$. Applying this to $(x^2)^3$, we get: $$ (x^2)^3 = x^{2 \cdot 3} = x^6 $$ So, the numerator becomes: $$ (x^2y)^3 = x^6y^3 $$
- Rewrite the expression with the simplified numerator
Now we rewrite the original expression with the simplified numerator: $$ \frac{(x^2y)^3}{xy^2} = \frac{x^6y^3}{xy^2} $$
- Apply the quotient of powers rule
The quotient of powers rule states that $\frac{a^m}{a^n} = a^{m-n}$. Applying this to the expression, we get: $$ \frac{x^6y^3}{xy^2} = \frac{x^6}{x^1} \cdot \frac{y^3}{y^2} = x^{6-1} \cdot y^{3-2} = x^5y^1 = x^5y $$
- Identify the correct option The simplified expression is $x^5y$, which corresponds to option A.
A. $x^5y$
More Information
The expression was simplified using the power of a product rule, power of a power rule, and quotient of powers rule. These rules are fundamental in simplifying expressions with exponents.
Tips
A common mistake is incorrectly applying the power of a product rule or the power of a power rule. For example, students might mistakenly calculate $(x^2)^3$ as $x^5$ instead of $x^6$. Another common mistake is forgetting to subtract the exponents correctly when dividing terms with the same base. Always double-check the application of the exponent rules.
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