Question - Simplified Expression by Dividing Numerical Coefficients and Variables with Exponents
Solution:
To simplify the given expression, divide the terms in the numerator by the terms in the denominator.The expression is:\[ \frac{7x^3y^{-5}}{21y^6} \]Divide the numerical coefficients:\[ \frac{7}{21} = \frac{1}{3} \]For the variables, use the laws of exponents to divide the terms with the same base:\[ x^{a} \div x^{b} = x^{a-b} \]\[ y^{a} \div y^{b} = y^{a-b} \]Applying the laws of exponents, you get:\[ x^3 \div x^0 = x^{3-0} = x^3 \](Any term to the power of 0 is 1, which is why the x term in the denominator is considered $$x^0$$)\[ y^{-5} \div y^6 = y^{-5-6} = y^{-11} \]Now put it all together:\[ \frac{1}{3} x^3 y^{-11} \]The negative exponent indicates the reciprocal:\[ y^{-11} = \frac{1}{y^{11}} \]Thus, the simplified expression is:\[ \frac{x^3}{3y^{11}} \]
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