Question - Simplification of an Algebraic Fraction with Negative Exponents
Solution:
$$\left( \frac{4a^6b^5c^{-2}}{(2a^{-4}b^6c)^3} \right)^2 = \left( \frac{4a^6b^5c^{-2}}{8a^{-12}b^{18}c^3} \right)^2 = \left( \frac{4 \cdot a^{6-(-12)} \cdot b^{5-18} \cdot c^{-2-3}}{8} \right)^2 = \left( \frac{4 \cdot a^{18} \cdot b^{-13} \cdot c^{-5}}{8} \right)^2 = \left( \frac{a^{18} \cdot b^{-13} \cdot c^{-5}}{2} \right)^2 = \frac{a^{36} \cdot b^{-26} \cdot c^{-10}}{4}$$
Tutor Solution:
To simplify the algebraic expression $$\frac{4a^6b^5c^{-2}}{(2a^{-4}b^6c)^3}$$, we will follow the properties of exponents step by step.First, let's handle the denominator. Since it is raised to the power of 3, we will apply the power rule $$(x^m)^n = x^{mn}$$:$$(2a^{-4}b^6c)^3 = 2^3(a^{-4})^3(b^6)^3c^3 = 8a^{-12}b^{18}c^3$$Now, write the initial expression with this expanded denominator:$$\frac{4a^6b^5c^{-2}}{8a^{-12}b^{18}c^3}$$Next, we apply the division rule of exponents $$x^m/x^n = x^{m-n}$$. Since we are dividing terms with the same base, we subtract the exponents:$$= 4a^{6 - (-12)}b^{5 - 18}c^{-2 - 3}$$$$= 4a^{6 + 12}b^{5 - 18}c^{-2 - 3}$$$$= 4a^{18}b^{-13}c^{-5}$$Lastly, we handle the coefficients and write the final expression without negative exponents by taking reciprocals for $$b$$ and $$c$$:$$\frac{4}{8}a^{18}b^{-13}c^{-5} = \frac{1}{2}a^{18}\frac{1}{b^{13}}\frac{1}{c^5}$$So the simplified expression is:$$\frac{1}{2}a^{18}b^{-13}c^{-5}$$Another way to present the final answer without negative exponents is by writing all terms in the numerator and denominator properly:$$\frac{a^{18}}{2b^{13}c^5}$$This is the fully simplified expression of the given algebraic fraction.
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