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How to Simplify the Rational Exponent Expression (18x^-3y^5)/(9x^9y^-4)
Mathematics
Grade 9 (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: $2x^{-12}y^{9}$. For a form without negative exponents, rewrite it as $\\frac{2y^{9}}{x^{12}}$ using the rule $a^{-n}=\\frac{1}{a^n}$
Knowledge Points Involved
1
Quotient Rule for Exponents
The 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 rule states that for any non-zero real number $a$ and positive integer $n$, $a^{-n}=\\frac{1}{a^n}$. It converts negative exponents into positive exponents by moving the base between the numerator and denominator of a fraction.
3
Simplification of Rational Expressions with Exponents
This involves combining coefficient simplification (dividing constants) and exponent rules to reduce complex rational expressions with variables and exponents to their simplest form, either with positive exponents or in a standard algebraic form.
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