Question

name:simplify each expression. write answers using positive1. $3x^{2}y \cdot 2x^{3}y$2. $(2a^{2}b)^{2}$3. $\frac{10m^{4}n^{2}}{5m^{2}n}$4. $(2x^{-1}y^{3})^{2}$5. $4p^{2}(2p^{-1})$6. $(3a^{2}b^{-1})^{2}$7. $\frac{6x^{3}y^{-1}}{3xy^{2}}$

Explanation:

1. Problem 1:

Step1: Group constants and like terms

$(3 \cdot 2) \cdot (x^2 \cdot x^3) \cdot (y \cdot y)$

Step2: Multiply constants, add exponents

$6 \cdot x^{2+3} \cdot y^{1+1} = 6x^5y^2$

2. Problem 2:

Step1: Apply power of a product rule

$2^2 \cdot (a^2)^2 \cdot b^2$

Step2: Calculate powers, multiply exponents

$4 \cdot a^{2 \cdot 2} \cdot b^2 = 4a^4b^2$

3. Problem 3:

Step1: Divide constants, subtract exponents

$\frac{10}{5} \cdot m^{4-2} \cdot n^{2-1}$

Step2: Simplify each term

$2 \cdot m^2 \cdot n^1 = 2m^2n$

4. Problem 4:

Step1: Apply power of a product rule

$2^2 \cdot (x^{-1})^2 \cdot (y^3)^2$

Step2: Calculate powers, multiply exponents

$4 \cdot x^{-1 \cdot 2} \cdot y^{3 \cdot 2} = 4x^{-2}y^6$

Step3: Rewrite with positive exponents

$\frac{4y^6}{x^2}$

5. Problem 5:

Step1: Group constants and like terms

$(4 \cdot 2) \cdot (p^2 \cdot p^{-1})$

Step2: Multiply constants, add exponents

$8 \cdot p^{2+(-1)} = 8p^1 = 8p$

6. Problem 6:

Step1: Apply power of a product rule

$3^2 \cdot (a^2)^2 \cdot (b^{-1})^2$

Step2: Calculate powers, multiply exponents

$9 \cdot a^{2 \cdot 2} \cdot b^{-1 \cdot 2} = 9a^4b^{-2}$

Step3: Rewrite with positive exponents

$\frac{9a^4}{b^2}$

7. Problem 7:

Step1: Divide constants, subtract exponents

$\frac{6}{3} \cdot x^{3-1} \cdot y^{-1-2}$

Step2: Simplify each term

$2 \cdot x^2 \cdot y^{-3}$

Step3: Rewrite with positive exponents

$\frac{2x^2}{y^3}$

Answer:

1. $6x^5y^2$
2. $4a^4b^2$
3. $2m^2n$
4. $\frac{4y^6}{x^2}$
5. $8p$
6. $\frac{9a^4}{b^2}$
7. $\frac{2x^2}{y^3}$