Solve (Bpiromega)^2/2Rquadtext{(b)}(Bpir^2omega)^2/8R

$$\frac { ( B \pi r \omega ) ^ { 2 } } { 2 R } \quad \text { (b) } \frac { ( B \pi r ^ { 2 } \omega ) ^ { 2 } } { 8 R }$$

Evaluate

$$\frac{br^{6}\left(\pi B\omega \right)^{4}}{16R^{2}}$$

Solution Steps

Expand \left(B\pi r\omega \right)^{2}.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b\times \frac{\left(B\pi r^{2}\omega \right)^{2}}{8R}$$

Expand \left(B\pi r^{2}\omega \right)^{2}.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b\times \frac{B^{2}\pi ^{2}\left(r^{2}\right)^{2}\omega ^{2}}{8R}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b\times \frac{B^{2}\pi ^{2}r^{4}\omega ^{2}}{8R}$$

Express \frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b as a single fraction.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}b}{2R}\times \frac{B^{2}\pi ^{2}r^{4}\omega ^{2}}{8R}$$

Multiply \frac{B^{2}\pi ^{2}r^{2}\omega ^{2}b}{2R} times \frac{B^{2}\pi ^{2}r^{4}\omega ^{2}}{8R} by multiplying numerator times numerator and denominator times denominator.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}bB^{2}\pi ^{2}r^{4}\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{2}r^{2}\omega ^{2}b\pi ^{2}r^{4}\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{4}r^{2}\omega ^{2}br^{4}\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{2}b\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{4}b}{2R\times 8R}$$

Multiply R and R to get R^{2}.

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{4}b}{2R^{2}\times 8}$$

Multiply 2 and 8 to get 16.

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{4}b}{16R^{2}}$$

Expand

$$\frac{br^{6}\left(\pi B\omega \right)^{4}}{16R^{2}}$$

Solution Steps

Expand \left(B\pi r\omega \right)^{2}.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b\times \frac{\left(B\pi r^{2}\omega \right)^{2}}{8R}$$

Expand \left(B\pi r^{2}\omega \right)^{2}.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b\times \frac{B^{2}\pi ^{2}\left(r^{2}\right)^{2}\omega ^{2}}{8R}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b\times \frac{B^{2}\pi ^{2}r^{4}\omega ^{2}}{8R}$$

Express \frac{B^{2}\pi ^{2}r^{2}\omega ^{2}}{2R}b as a single fraction.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}b}{2R}\times \frac{B^{2}\pi ^{2}r^{4}\omega ^{2}}{8R}$$

Multiply \frac{B^{2}\pi ^{2}r^{2}\omega ^{2}b}{2R} times \frac{B^{2}\pi ^{2}r^{4}\omega ^{2}}{8R} by multiplying numerator times numerator and denominator times denominator.

$$\frac{B^{2}\pi ^{2}r^{2}\omega ^{2}bB^{2}\pi ^{2}r^{4}\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{2}r^{2}\omega ^{2}b\pi ^{2}r^{4}\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{4}r^{2}\omega ^{2}br^{4}\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{2}b\omega ^{2}}{2R\times 8R}$$

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

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{4}b}{2R\times 8R}$$

Multiply R and R to get R^{2}.

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{4}b}{2R^{2}\times 8}$$

Multiply 2 and 8 to get 16.

$$\frac{B^{4}\pi ^{4}r^{6}\omega ^{4}b}{16R^{2}}$$