Question

bellwork 1/21
1.) $7x^{-3}y^{4} \cdot 3y^{-2} \cdot 2xy$
2.) $(3m^{-2}n^{4})^{3}$
3.) $\frac{3m^{-2}n^{3}p^{-7}}{6m^{2}n^{-1}p^{3}}$

Explanation:

Step1: Multiply constant coefficients

$7 \times 3 \times 2 = 42$

Step2: Combine $x$-terms via exponent addition

$x^{-3} \times x^1 = x^{-3+1} = x^{-2} = \frac{1}{x^2}$

Step3: Combine $y$-terms via exponent addition

$y^4 \times y^{-2} \times y^1 = y^{4-2+1} = y^3$

Step4: Multiply all simplified terms

$42 \times \frac{1}{x^2} \times y^3 = \frac{42y^3}{x^2}$

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Step1: Distribute exponent to constant

$3^3 = 27$

Step2: Distribute exponent to $m$-term

$(m^{-2})^3 = m^{-2 \times 3} = m^{-6} = \frac{1}{m^6}$

Step3: Distribute exponent to $n$-term

$(n^4)^3 = n^{4 \times 3} = n^{12}$

Step4: Multiply all simplified terms

$27 \times \frac{1}{m^6} \times n^{12} = \frac{27n^{12}}{m^6}$

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Step1: Simplify constant coefficients

$\frac{3}{6} = \frac{1}{2}$

Step2: Combine $m$-terms via exponent subtraction

$m^{-2} \div m^2 = m^{-2-2} = m^{-4} = \frac{1}{m^4}$

Step3: Combine $n$-terms via exponent subtraction

$n^3 \div n^{-1} = n^{3-(-1)} = n^{4}$

Step4: Combine $p$-terms via exponent subtraction

$p^{-7} \div p^3 = p^{-7-3} = p^{-10} = \frac{1}{p^{10}}$

Step5: Multiply all simplified terms

$\frac{1}{2} \times \frac{1}{m^4} \times n^4 \times \frac{1}{p^{10}} = \frac{n^4}{2m^4p^{10}}$

Answer:

1. $\boldsymbol{\frac{42y^3}{x^2}}$
2. $\boldsymbol{\frac{27n^{12}}{m^6}}$
3. $\boldsymbol{\frac{n^4}{2m^4p^{10}}}$