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Chapter 2: Problem 24

Simplify. \(\frac{\left(-3 a^{4}\right)^{3}}{(6 b)^{2}}\)

Short Answer

Expert verified
The simplified expression is \(-\frac{9a^{12}}{12b^{2}}\).

Step by step solution

01

Rewrite the expression with proper brackets and signs

Rewrite the given expression by applying the parenthesis and negative signs properly. \(\frac{\left(-3 a^{4}\right)^{3}}{(6 b)^{2}} = \frac{(-1)^{3}(3)^{3}(a^{4})^{3}}{(6^{2})(b^{2})}\)
02

Apply the exponent rules

Now apply the exponent rules to simplify the expression: \(\frac{(-1)^{3}(3)^{3}(a^{4})^{3}}{(6^{2})(b^{2})} =\frac{-1\cdot27\cdot a^{12}}{36\cdot b^{2}}\) Note that we multiplied the exponents in the case of \((a^{4})^{3}\) giving us \(a^{12}\), and we have \(-1\) and \(27\) as results for \((-1)^{3}\) and \((3)^{3}\) respectively.
03

Simplify and reduce

Simplify and reduce the expression further: \(\frac{-1\cdot27\cdot a^{12}}{36\cdot b^{2}} = \frac{-27a^{12}}{36b^{2}}= \frac{-9\cdot3a^{12}}{12\cdot3b^{2}}\) Now divide both numerator and denominator by the greatest common divisor, which is \(3\): \(\frac{-9\cdot3a^{12}}{12\cdot3b^{2}} = \frac{-9a^{12}}{12b^{2}}\)
04

Rewrite the simplified expression

Finally, rewrite the simplified expression in the proper format: \(-\frac{9a^{12}}{12b^{2}}\) The simplified expression is: \(-\frac{9a^{12}}{12b^{2}}\)

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