Solve g^{-1}h^-1*g^-1h^0*g^8h

$$g ^ { - 1 } h ^ { - 1 } \cdot g ^ { - 1 } h ^ { 0 } \cdot g ^ { 8 } h$$

Evaluate

$$g^{6}$$

Steps Using Commutative Property of Multiplication

Use the rules of exponents to simplify the expression.

$$\frac{1}{g}\times \frac{1}{h}\times \frac{1}{g}h^{0}h^{1}g^{8}$$

Use the Commutative Property of Multiplication.

$$\frac{1}{g}h^{0}\times \frac{1}{h}h^{1}\times \frac{1}{g}g^{8}$$

To multiply powers of the same base, add their exponents.

$$\frac{1}{g}h^{0}h^{-1+1}g^{-1+8}$$

Add the exponents -1 and 1.

$$\frac{1}{g}h^{0}h^{0}g^{-1+8}$$

For any number a except 0, a^{0}=1.

$$\frac{1}{g}h^{0}g^{-1+8}$$

Add the exponents -1 and 8.

$$\frac{1}{g}h^{0}g^{7}$$

Raise h to the power 0.

$$\frac{1}{g}g^{7}$$

Differentiate w.r.t. g

$$6g^{5}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}g}(g^{-2}h^{-1}h^{0}g^{8}h)$$

To multiply powers of the same base, add their exponents. Add -2 and 8 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}g}(g^{6}h^{-1}h^{0}h)$$

To multiply powers of the same base, add their exponents. Add -1 and 0 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}g}(g^{6}h^{-1}h)$$

Multiply h^{-1} and h to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}g}(g^{6})$$

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

$$6g^{6-1}$$

Subtract 1 from 6.

$$6g^{5}$$