Solve frac{((b^{-3})^{2})^{-1}}{b^{-3^{2}}(b^{-3})^{2}*b^{(-3)^{2}}}*{{b*b^{2}(b^-12)^-2}^-1}^-4

$$\frac { ( ( b ^ { - 3 } ) ^ { 2 } ) ^ { - 1 } } { b ^ { - 3 ^ { 2 } } ( b ^ { - 3 } ) ^ { 2 } \cdot b ^ { ( - 3 ) ^ { 2 } } } \cdot \{ \{ b \cdot b ^ { 2 } ( b ^ { - 12 } ) ^ { - 2 } \} ^ { - 1 } \} ^ { - 4 }$$

Evaluate

$$b^{120}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{\left(b^{-6}\right)^{-1}}{b^{-3^{2}}\left(b^{-3}\right)^{2}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}\left(b^{-12}\right)^{-2}\right)^{-1}\right)^{-4}$$

To raise a power to another power, multiply the exponents. Multiply -6 and -1 to get 6.

$$\frac{b^{6}}{b^{-3^{2}}\left(b^{-3}\right)^{2}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}\left(b^{-12}\right)^{-2}\right)^{-1}\right)^{-4}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}\left(b^{-12}\right)^{-2}\right)^{-1}\right)^{-4}$$

To raise a power to another power, multiply the exponents. Multiply -12 and -2 to get 24.

$$\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}b^{24}\right)^{-1}\right)^{-4}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(b^{3}b^{24}\right)^{-1}\right)^{-4}$$

To multiply powers of the same base, add their exponents. Add 3 and 24 to get 27.

$$\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(b^{27}\right)^{-1}\right)^{-4}$$

To raise a power to another power, multiply the exponents. Multiply 27 and -1 to get -27.

$$\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(b^{-27}\right)^{-4}$$

To raise a power to another power, multiply the exponents. Multiply -27 and -4 to get 108.

$$\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}b^{108}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{b^{12}}{b^{-3^{2}}b^{\left(-3\right)^{2}}}b^{108}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{b^{12}}{b^{-9}b^{\left(-3\right)^{2}}}b^{108}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{b^{12}}{b^{-9}b^{9}}b^{108}$$

Multiply b^{-9} and b^{9} to get 1.

$$\frac{b^{12}}{1}b^{108}$$

Anything divided by one gives itself.

$$b^{12}b^{108}$$

To multiply powers of the same base, add their exponents. Add 12 and 108 to get 120.

$$b^{120}$$

Differentiate w.r.t. b

$$120b^{119}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\left(b^{-6}\right)^{-1}}{b^{-3^{2}}\left(b^{-3}\right)^{2}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}\left(b^{-12}\right)^{-2}\right)^{-1}\right)^{-4})$$

To raise a power to another power, multiply the exponents. Multiply -6 and -1 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{6}}{b^{-3^{2}}\left(b^{-3}\right)^{2}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}\left(b^{-12}\right)^{-2}\right)^{-1}\right)^{-4})$$

To raise a power to another power, multiply the exponents. Multiply -3 and 2 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}\left(b^{-12}\right)^{-2}\right)^{-1}\right)^{-4})$$

To raise a power to another power, multiply the exponents. Multiply -12 and -2 to get 24.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(bb^{2}b^{24}\right)^{-1}\right)^{-4})$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(b^{3}b^{24}\right)^{-1}\right)^{-4})$$

To multiply powers of the same base, add their exponents. Add 3 and 24 to get 27.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(\left(b^{27}\right)^{-1}\right)^{-4})$$

To raise a power to another power, multiply the exponents. Multiply 27 and -1 to get -27.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}\left(b^{-27}\right)^{-4})$$

To raise a power to another power, multiply the exponents. Multiply -27 and -4 to get 108.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{6}}{b^{-3^{2}}b^{-6}b^{\left(-3\right)^{2}}}b^{108})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{12}}{b^{-3^{2}}b^{\left(-3\right)^{2}}}b^{108})$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{12}}{b^{-9}b^{\left(-3\right)^{2}}}b^{108})$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{12}}{b^{-9}b^{9}}b^{108})$$

Multiply b^{-9} and b^{9} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{12}}{1}b^{108})$$

Anything divided by one gives itself.

$$\frac{\mathrm{d}}{\mathrm{d}b}(b^{12}b^{108})$$

To multiply powers of the same base, add their exponents. Add 12 and 108 to get 120.

$$\frac{\mathrm{d}}{\mathrm{d}b}(b^{120})$$

The derivative of ax^{n} is nax^{n-1}.

$$120b^{120-1}$$

Subtract 1 from 120.

$$120b^{119}$$