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Simplify the Exponential Rational Expression: $\\left( \\frac{3m^{-5}n^{2}}{4m^{-2}n^{0}} \\right)^{2} \\cdot \\left( \\frac{mn^{4}}{9n} \\right)^{2}$
Mathematics
Grade 9 (Junior High School)
Question Content
Simplify the expression: $\left( \frac{3m^{-5}n^{2}}{4m^{-2}n^{0}} \right)^{2} \cdot \left( \frac{mn^{4}}{9n} \right)^{2}$
Correct Answer
$\\frac{n^{10}}{144m^{8}}$
Detailed Solution Steps
1
Step 1: Simplify the terms inside each parentheses first using exponent rules. For $n^0$, recall any non-zero number to the power of 0 is 1, so $n^0=1$. For the first fraction: $\\frac{3m^{-5}n^{2}}{4m^{-2} \\cdot 1} = \\frac{3}{4}m^{-5-(-2)}n^{2} = \\frac{3}{4}m^{-3}n^{2}$. For the second fraction: $\\frac{mn^{4}}{9n} = \\frac{1}{9}m^{1}n^{4-1} = \\frac{1}{9}mn^{3}$
2
Step 2: Apply the power of a power rule $(a^b)^c=a^{b \\cdot c}$ to each squared fraction. For the first squared term: $\\left(\\frac{3}{4}m^{-3}n^{2}\\right)^2 = \\left(\\frac{3}{4}\\right)^2m^{-3 \\cdot 2}n^{2 \\cdot 2} = \\frac{9}{16}m^{-6}n^{4}$. For the second squared term: $\\left(\\frac{1}{9}mn^{3}\\right)^2 = \\left(\\frac{1}{9}\\right)^2m^{1 \\cdot 2}n^{3 \\cdot 2} = \\frac{1}{81}m^{2}n^{6}$
3
Step 3: Multiply the two simplified terms together. Multiply the coefficients first: $\\frac{9}{16} \\cdot \\frac{1}{81} = \\frac{1}{144}$. Then combine the $m$ terms using the product rule $a^b \\cdot a^c = a^{b+c}$: $m^{-6} \\cdot m^{2} = m^{-6+2}=m^{-4}$. Combine the $n$ terms: $n^{4} \\cdot n^{6}=n^{4+6}=n^{10}$
4
Step 4: Rewrite negative exponents as positive exponents in the denominator: $m^{-4} = \\frac{1}{m^4}$. Combine all parts to get the final simplified expression: $\\frac{1}{144} \\cdot \\frac{1}{m^4} \\cdot n^{10} = \\frac{n^{10}}{144m^{8}}$
Knowledge Points Involved
1
Zero Exponent Rule
The rule states that for any non-zero real number $a$, $a^0 = 1$. It applies when a non-zero base is raised to the power of 0, simplifying constant terms in rational expressions.
2
Quotient of Powers Rule
For any non-zero real number $a$ and integers $b$ and $c$, $\\frac{a^b}{a^c}=a^{b-c}$. This is used to simplify fractions with the same base by subtracting the exponent in the denominator from the exponent in the numerator.
3
Power of a Power Rule
For any non-zero real number $a$ and integers $b$ and $c$, $(a^b)^c=a^{b \\cdot c}$. This rule is applied when raising an exponential term to another power, by multiplying the exponents together.
4
Product of Powers Rule
For any non-zero real number $a$ and integers $b$ and $c$, $a^b \\cdot a^c=a^{b+c}$. It is used to multiply two exponential terms with the same base by adding their exponents.
5
Negative Exponent Rule
For any non-zero real number $a$ and positive integer $b$, $a^{-b}=\\frac{1}{a^b}$. This converts negative exponents to positive exponents by moving the term to the opposite part of the fraction (numerator to denominator or vice versa).
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