Solve x^2quadfrac{sqrt{3}}{2}x^2quadtext{(c)}x^2/sqrt{3}quadtext{(d)}x^2/2

$$x ^ { 2 } \quad \frac { \sqrt { 3 } } { 2 } x ^ { 2 } \quad \text { (c) } \frac { x ^ { 2 } } { \sqrt { 3 } } \quad \text { (d) } \frac { x ^ { 2 } } { 2 }$$

Evaluate

$$\frac{cdx^{8}}{4}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{4}\times \frac{\sqrt{3}}{2}c\times \frac{x^{2}}{\sqrt{3}}d\times \frac{x^{2}}{2}$$

Rationalize the denominator of \frac{x^{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$x^{4}\times \frac{\sqrt{3}}{2}c\times \frac{x^{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}d\times \frac{x^{2}}{2}$$

The square of \sqrt{3} is 3.

$$x^{4}\times \frac{\sqrt{3}}{2}c\times \frac{x^{2}\sqrt{3}}{3}d\times \frac{x^{2}}{2}$$

Express x^{4}\times \frac{\sqrt{3}}{2} as a single fraction.

$$\frac{x^{4}\sqrt{3}}{2}c\times \frac{x^{2}\sqrt{3}}{3}d\times \frac{x^{2}}{2}$$

Express \frac{x^{4}\sqrt{3}}{2}c as a single fraction.

$$\frac{x^{4}\sqrt{3}c}{2}\times \frac{x^{2}\sqrt{3}}{3}d\times \frac{x^{2}}{2}$$

Multiply \frac{x^{4}\sqrt{3}c}{2} times \frac{x^{2}\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{4}\sqrt{3}cx^{2}\sqrt{3}}{2\times 3}d\times \frac{x^{2}}{2}$$

Express \frac{x^{4}\sqrt{3}cx^{2}\sqrt{3}}{2\times 3}d as a single fraction.

$$\frac{x^{4}\sqrt{3}cx^{2}\sqrt{3}d}{2\times 3}\times \frac{x^{2}}{2}$$

Multiply \frac{x^{4}\sqrt{3}cx^{2}\sqrt{3}d}{2\times 3} times \frac{x^{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{4}\sqrt{3}cx^{2}\sqrt{3}dx^{2}}{2\times 3\times 2}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{x^{6}\sqrt{3}c\sqrt{3}dx^{2}}{2\times 3\times 2}$$

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

$$\frac{x^{8}\sqrt{3}c\sqrt{3}d}{2\times 3\times 2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{x^{8}\times 3cd}{2\times 3\times 2}$$

Cancel out 3 in both numerator and denominator.

$$\frac{cdx^{8}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{cdx^{8}}{4}$$

Differentiate w.r.t. x

$$2cdx^{7}$$