Question

1. simplify following expressions

a. \\(\frac{(3a^2)^3}{6a^{10}b^{-3}}\\)
b. \\((5x^2y^{-3})^0(-2x^{-3}y^4)\\)
c. \\(\frac{4×10^2}{2×10^{-3}}\\)

Explanation:

Step1: Simplify numerator using power rule

$(3a^2)^3 = 3^3 \cdot (a^2)^3 = 27a^6$

Step2: Rewrite negative exponent as positive

$\frac{27a^6}{6a^{10}b^{-3}} = \frac{27a^6b^3}{6a^{10}}$

Step3: Simplify coefficients and exponents

$\frac{27}{6} \cdot a^{6-10} \cdot b^3 = \frac{9}{2}a^{-4}b^3 = \frac{9b^3}{2a^4}$

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Step1: Apply zero exponent rule

$(5x^2y^{-3})^0 = 1$ (any non-zero term to 0 power is 1)

Step2: Multiply by remaining term

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

Step3: Rewrite negative exponent as positive

$-2x^{-3}y^4 = \frac{-2y^4}{x^3}$

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Step1: Split coefficient and power terms

$\frac{4 \times 10^3}{2 \times 10^{-3}} = \frac{4}{2} \times \frac{10^3}{10^{-3}}$

Step2: Simplify coefficients and exponents

$2 \times 10^{3-(-3)} = 2 \times 10^6$

Answer:

a. $\boldsymbol{\frac{9b^3}{2a^4}}$
b. $\boldsymbol{\frac{-2y^4}{x^3}}$
c. $\boldsymbol{2 \times 10^6}$