Solve [frac{(f^{3}g^{-8}h)^{7}}{(g^5h^-3f)^-8}]^5

$$[ \frac { ( f ^ { 3 } g ^ { - 8 } h ) ^ { 7 } } { ( g ^ { 5 } h ^ { - 3 } f ) ^ { - 8 } } ] ^ { 5 }$$

Evaluate

$$\frac{f^{145}}{g^{80}h^{85}}$$

Solution Steps

Expand \left(f^{3}g^{-8}h\right)^{7}.

$$\left(\frac{\left(f^{3}\right)^{7}\left(g^{-8}\right)^{7}h^{7}}{\left(g^{5}h^{-3}f\right)^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 7 to get 21.

$$\left(\frac{f^{21}\left(g^{-8}\right)^{7}h^{7}}{\left(g^{5}h^{-3}f\right)^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply -8 and 7 to get -56.

$$\left(\frac{f^{21}g^{-56}h^{7}}{\left(g^{5}h^{-3}f\right)^{-8}}\right)^{5}$$

Expand \left(g^{5}h^{-3}f\right)^{-8}.

$$\left(\frac{f^{21}g^{-56}h^{7}}{\left(g^{5}\right)^{-8}\left(h^{-3}\right)^{-8}f^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -8 to get -40.

$$\left(\frac{f^{21}g^{-56}h^{7}}{g^{-40}\left(h^{-3}\right)^{-8}f^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -8 to get 24.

$$\left(\frac{f^{21}g^{-56}h^{7}}{g^{-40}h^{24}f^{-8}}\right)^{5}$$

Cancel out h^{7} in both numerator and denominator.

$$\left(\frac{g^{-56}f^{21}}{g^{-40}f^{-8}h^{17}}\right)^{5}$$

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

$$\left(\frac{g^{-56}f^{29}}{g^{-40}h^{17}}\right)^{5}$$

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

$$\left(\frac{f^{29}}{g^{16}h^{17}}\right)^{5}$$

To raise \frac{f^{29}}{g^{16}h^{17}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(f^{29}\right)^{5}}{\left(g^{16}h^{17}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 29 and 5 to get 145.

$$\frac{f^{145}}{\left(g^{16}h^{17}\right)^{5}}$$

Expand \left(g^{16}h^{17}\right)^{5}.

$$\frac{f^{145}}{\left(g^{16}\right)^{5}\left(h^{17}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 16 and 5 to get 80.

$$\frac{f^{145}}{g^{80}\left(h^{17}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 17 and 5 to get 85.

$$\frac{f^{145}}{g^{80}h^{85}}$$

Expand

$$\frac{f^{145}}{g^{80}h^{85}}$$

Solution Steps

Expand \left(f^{3}g^{-8}h\right)^{7}.

$$\left(\frac{\left(f^{3}\right)^{7}\left(g^{-8}\right)^{7}h^{7}}{\left(g^{5}h^{-3}f\right)^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 7 to get 21.

$$\left(\frac{f^{21}\left(g^{-8}\right)^{7}h^{7}}{\left(g^{5}h^{-3}f\right)^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply -8 and 7 to get -56.

$$\left(\frac{f^{21}g^{-56}h^{7}}{\left(g^{5}h^{-3}f\right)^{-8}}\right)^{5}$$

Expand \left(g^{5}h^{-3}f\right)^{-8}.

$$\left(\frac{f^{21}g^{-56}h^{7}}{\left(g^{5}\right)^{-8}\left(h^{-3}\right)^{-8}f^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -8 to get -40.

$$\left(\frac{f^{21}g^{-56}h^{7}}{g^{-40}\left(h^{-3}\right)^{-8}f^{-8}}\right)^{5}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -8 to get 24.

$$\left(\frac{f^{21}g^{-56}h^{7}}{g^{-40}h^{24}f^{-8}}\right)^{5}$$

Cancel out h^{7} in both numerator and denominator.

$$\left(\frac{g^{-56}f^{21}}{g^{-40}f^{-8}h^{17}}\right)^{5}$$

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

$$\left(\frac{g^{-56}f^{29}}{g^{-40}h^{17}}\right)^{5}$$

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

$$\left(\frac{f^{29}}{g^{16}h^{17}}\right)^{5}$$

To raise \frac{f^{29}}{g^{16}h^{17}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(f^{29}\right)^{5}}{\left(g^{16}h^{17}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 29 and 5 to get 145.

$$\frac{f^{145}}{\left(g^{16}h^{17}\right)^{5}}$$

Expand \left(g^{16}h^{17}\right)^{5}.

$$\frac{f^{145}}{\left(g^{16}\right)^{5}\left(h^{17}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 16 and 5 to get 80.

$$\frac{f^{145}}{g^{80}\left(h^{17}\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 17 and 5 to get 85.

$$\frac{f^{145}}{g^{80}h^{85}}$$