Question

sitive exponents.

1. \\(\frac{3x^{-4}y^{-3}}{4x^{-4}y^{0}}\\)
2. \\(\frac{3x^{-2}y^{-4}}{x}\\)
3. \\(\frac{2x^{2}}{3x^{2}y^{-1} \cdot 2x^{-2}y^{4}}\\)
4. \\(\frac{4m^{-3}n^{2} \cdot 2m^{-1}n^{3}}{3m^{-2}n^{3}}\\)

Explanation:

Problem 2: Simplify $\boldsymbol{\frac{3x^{-4}y^{-3}}{4x^{-4}y^{0}}}$

Step 1: Simplify \(x\)-exponents

Use the rule \(\frac{a^m}{a^n}=a^{m - n}\). For \(x\)-terms: \(x^{-4}/x^{-4}=x^{-4-(-4)} = x^{0}=1\) (since \(a^0 = 1\) for \(a
eq0\)).

Step 2: Simplify \(y\)-exponents

For \(y\)-terms: \(y^{-3}/y^{0}=y^{-3 - 0}=y^{-3}=\frac{1}{y^{3}}\) (using \(a^{-n}=\frac{1}{a^{n}}\)).

Step 3: Simplify constants and combine

The constant term is \(3/4\), and we multiply by the results from \(x\) and \(y\) terms: \(\frac{3}{4}\times1\times\frac{1}{y^{3}}=\frac{3}{4y^{3}}\).

Step 1: Simplify \(x\)-exponents

Rewrite \(x\) as \(x^{1}\). Use \(\frac{a^m}{a^n}=a^{m - n}\): \(x^{-2}/x^{1}=x^{-2 - 1}=x^{-3}=\frac{1}{x^{3}}\).

Step 2: Combine with \(y\) and constant

The \(y\)-term is \(y^{-4}=\frac{1}{y^{4}}\) and the constant is \(3\). Multiply them: \(3\times\frac{1}{x^{3}}\times\frac{1}{y^{4}}=\frac{3}{x^{3}y^{4}}\).

Step 1: Simplify denominator (multiply \(x\) and \(y\) terms)

For \(x\)-terms in denominator: \(x^{2}\cdot x^{-2}=x^{2+(-2)} = x^{0}=1\). For \(y\)-terms: \(y^{-1}\cdot y^{4}=y^{-1 + 4}=y^{3}\). For constants: \(3\times2 = 6\). So denominator becomes \(6\times1\times y^{3}=6y^{3}\).

Step 2: Simplify the fraction

Now we have \(\frac{2x^{2}}{6y^{3}}\). Simplify the constant fraction \(2/6=\frac{1}{3}\), so we get \(\frac{x^{2}}{3y^{3}}\).

Answer:

\(\frac{3}{4y^{3}}\)

Problem 4: Simplify $\boldsymbol{\frac{3x^{-2}y^{-4}}{x}}$