Math Problem Statement

Simplify $\left(\frac{3m^{-5}n^2}{4m^{-2}n^0}\right)^2\cdot\left(\frac{mn^4}{9n}\right)^2$

Solution

Let's simplify the given expression step by step:

(3m5n24m2n0)2(mn49n)2\left(\frac{3m^{-5}n^2}{4m^{-2}n^0}\right)^2\cdot\left(\frac{mn^4}{9n}\right)^2

Step 1: Simplify Each Fraction Inside the Parentheses

First Fraction:

3m5n24m2n0\frac{3m^{-5}n^2}{4m^{-2}n^0}

  • Since n0=1n^0 = 1, we ignore it.
  • Divide the powers of mm:

m5÷m2=m5(2)=m3m^{-5} \div m^{-2} = m^{-5 - (-2)} = m^{-3}

Thus, the first fraction simplifies to:

3m3n24\frac{3m^{-3}n^2}{4}

Second Fraction:

mn49n\frac{mn^4}{9n}

  • Divide the powers of nn:

n4÷n=n41=n3n^4 \div n = n^{4-1} = n^3

Thus, the second fraction simplifies to:

mn39\frac{mn^3}{9}

Step 2: Square Each Simplified Fraction

First Squared Expression:

(3m3n24)2\left(\frac{3m^{-3}n^2}{4}\right)^2

Squaring each term:

32m6n442=9m6n416\frac{3^2 m^{-6} n^4}{4^2} = \frac{9m^{-6}n^4}{16}

Second Squared Expression:

(mn39)2\left(\frac{mn^3}{9}\right)^2

Squaring each term:

m2n681\frac{m^2 n^6}{81}

Step 3: Multiply the Two Results

(9m6n416)(m2n681)\left(\frac{9m^{-6}n^4}{16}\right) \cdot \left(\frac{m^2 n^6}{81}\right)

Multiply the numerators:

9m6n4×m2n6=9m6+2n4+6=9m4n109m^{-6}n^4 \times m^2 n^6 = 9m^{-6+2}n^{4+6} = 9m^{-4}n^{10}

Multiply the denominators:

16×81=129616 \times 81 = 1296

Thus, the expression simplifies to:

9m4n101296\frac{9m^{-4}n^{10}}{1296}

Step 4: Simplify the Coefficient

91296=1144\frac{9}{1296} = \frac{1}{144}

Thus, the final simplified expression is:

m4n10144\frac{m^{-4}n^{10}}{144}

Or, writing with positive exponents:

n10144m4\frac{n^{10}}{144m^4}

Final Answer:

n10144m4\frac{n^{10}}{144m^4}

Would you like me to clarify any step or go into more details?

Related Questions:

  1. How do you simplify expressions with negative exponents?
  2. What are the laws of exponents and how do they apply to fraction simplifications?
  3. How do you handle complex fractions with variables in exponent form?
  4. What is the process of rationalizing expressions with negative exponents?
  5. How can exponent rules be applied in algebraic fraction simplifications?

Tip: Always simplify inside parentheses before applying exponent rules!

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponent Rules
Fraction Simplification

Formulas

a^m / a^n = a^{m-n}
(a/b)^n = a^n / b^n
a^m * a^n = a^{m+n}

Theorems

Laws of Exponents
Fraction Multiplication

Suitable Grade Level

Grades 8-10