Solve 21z^10/7z^9

$$\frac { 21 z ^ { 10 } } { 7 z ^ { 9 } }$$

Evaluate

$$3z$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(21z^{10}\right)^{1}\times \frac{1}{7z^{9}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$21^{1}\left(z^{10}\right)^{1}\times \frac{1}{7}\times \frac{1}{z^{9}}$$

Use the Commutative Property of Multiplication.

$$21^{1}\times \frac{1}{7}\left(z^{10}\right)^{1}\times \frac{1}{z^{9}}$$

To raise a power to another power, multiply the exponents.

$$21^{1}\times \frac{1}{7}z^{10}z^{9\left(-1\right)}$$

Multiply 9 times -1.

$$21^{1}\times \frac{1}{7}z^{10}z^{-9}$$

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

$$21^{1}\times \frac{1}{7}z^{10-9}$$

Add the exponents 10 and -9.

$$21^{1}\times \frac{1}{7}z^{1}$$

Raise 21 to the power 1.

$$21\times \frac{1}{7}z^{1}$$

Multiply 21 times \frac{1}{7}.

$$3z^{1}$$

For any term t, t^{1}=t.

$$3z$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{21^{1}z^{10}}{7^{1}z^{9}}$$

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

$$\frac{21^{1}z^{10-9}}{7^{1}}$$

Subtract 9 from 10.

$$\frac{21^{1}z^{1}}{7^{1}}$$

Divide 21 by 7.

$$3z^{1}$$

For any term t, t^{1}=t.

$$3z$$

Differentiate w.r.t. z

$$3$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{21}{7}z^{10-9})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(3z^{1})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3z^{1-1}$$

Do the arithmetic.

$$3z^{0}$$

For any term t except 0, t^{0}=1.

$$3\times 1$$

For any term t, t\times 1=t and 1t=t.

$$3$$