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How to Simplify the Rational Exponential Expression $\\left(\\frac{18x^{-3}y^{5}}{9x^{9}y^{-4}}\\right)$
Mathematics
Grade 9 of Junior High School
Question Content
Simplify the expression: $\left(\frac{18x^{-3}y^{5}}{9x^{9}y^{-4}}\right)$
Correct Answer
$2x^{-12}y^{9}$ or $\frac{2y^{9}}{x^{12}}$
Detailed Solution Steps
1
Step 1: Simplify the constant coefficients first. Divide 18 by 9, which equals 2, so the expression becomes $\frac{2x^{-3}y^{5}}{x^{9}y^{-4}}$
2
Step 2: Apply the quotient rule for exponents ($\\frac{a^m}{a^n}=a^{m-n}$) to the $x$-terms. Calculate $x^{-3-9}=x^{-12}$
3
Step 3: Apply the quotient rule for exponents to the $y$-terms. Calculate $y^{5-(-4)}=y^{5+4}=y^{9}$
4
Step 4: Combine all simplified parts to get the final result: $2x^{-12}y^{9}$. This can also be rewritten with positive exponents as $\frac{2y^{9}}{x^{12}}$
Knowledge Points Involved
1
Quotient Rule for Exponents
The quotient rule states that for any non-zero real number $a$, and integers $m$ and $n$, $\\frac{a^m}{a^n}=a^{m-n}$. It is used to simplify expressions where the same base is divided, by subtracting the exponent of the denominator from the exponent of the numerator.
2
Negative Exponent Rule
The negative exponent rule states that $a^{-n}=\\frac{1}{a^n}$ for any non-zero real number $a$ and positive integer $n$. This rule is used to rewrite terms with negative exponents into equivalent expressions with positive exponents, which is often the preferred form for simplified results.
3
Simplification of Rational Expressions with Exponents
This involves simplifying coefficients by basic division, then applying exponent rules to like bases separately. The goal is to rewrite the expression with the fewest terms and positive exponents when possible.
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