# How do you multiply and simplify \frac { 6b } { 4b ^ { 2} } \cdot \frac { 8b ^ { 4} } { 9b ^ { 5} } ?

Jun 15, 2017

See a solution process below:

#### Explanation:

First, rewrite this expression as:

$\frac{6 \cdot 8}{4 \cdot 9} \cdot \frac{b \cdot {b}^{4}}{{b}^{2} \cdot {b}^{5}}$

Next, factor and cancel common terms in the contants:

$\frac{3 \cdot 2 \cdot 4 \cdot 2}{4 \cdot 3 \cdot 3} \cdot \frac{b \cdot {b}^{4}}{{b}^{2} \cdot {b}^{5}} \implies$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \cdot 2 \cdot \textcolor{b l u e}{\cancel{\textcolor{b l a c k}{4}}} \cdot 2}{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{4}}} \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \cdot 3} \cdot \frac{b \cdot {b}^{4}}{{b}^{2} \cdot {b}^{5}} \implies$

$\frac{2 \cdot 2}{3} \cdot \frac{b \cdot {b}^{4}}{{b}^{2} \cdot {b}^{5}} \implies$

$\frac{4}{3} \cdot \frac{b \cdot {b}^{4}}{{b}^{2} \cdot {b}^{5}}$

Now, use these rules of exponents to simplify the $b$ terms in the numerator:

$a = {a}^{\textcolor{red}{1}}$ and ${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$

$\frac{4}{3} \cdot \frac{{b}^{\textcolor{red}{1}} \times {b}^{\textcolor{b l u e}{4}} \cdot {b}^{4}}{{b}^{2} \cdot {b}^{5}} \implies \frac{4}{3} \cdot \frac{{b}^{\textcolor{red}{1} + \textcolor{b l u e}{4}}}{{b}^{2} \cdot {b}^{5}} \implies \frac{4}{3} \cdot {b}^{5} / \left({b}^{2} \cdot {b}^{5}\right) \implies$

$\frac{4}{3} \cdot \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{{b}^{5}}}}}{{b}^{2} \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{{b}^{5}}}}} \implies$

$\frac{4}{3 {b}^{2}}$

Jun 15, 2017

$\frac{6 b}{4 {b}^{2}} \cdot \frac{8 {b}^{4}}{9 {b}^{5}}$ simplifies to color(blue)(4/(3b^2).

#### Explanation:

Multiply and simplify.

$\frac{6 b}{4 {b}^{2}} \cdot \frac{8 {b}^{4}}{9 {b}^{5}}$

Multiply the numerators and denominators across the two fractions.

$\frac{6 b \times 8 {b}^{4}}{4 {b}^{2} \times 9 {b}^{5}}$

Multiply the coefficients.

$\frac{6 \times 8 \times b \times {b}^{4}}{4 \times 9 \times {b}^{2} \times {b}^{5}}$

Simplify.

$\frac{48 \times b \times {b}^{4}}{36 \times {b}^{2} \times {b}^{5}}$

Apply exponent product rule: ${x}^{m} {x}^{n} = {x}^{m + n}$.

$\frac{48 \times {b}^{1 + 4}}{36 \times {b}^{2 + 5}}$

Simplify.

$\frac{48 {b}^{5}}{36 {b}^{7}}$

Simplify $\frac{48}{36}$.

$\frac{\left(48 \div 12\right) {b}^{5}}{\left(36 \div 12\right) {b}^{7}}$

$\frac{4 {b}^{5}}{3 {b}^{7}}$

Apply quotient exponent rule: ${x}^{m} / {x}^{n} = {x}^{m - n}$.

$\frac{4 {b}^{5 - 7}}{3}$

Simplify.

$\frac{4 {b}^{- 2}}{3}$

Apply the negative exponent rule: ${x}^{- m} = \frac{1}{x} ^ m$.

$\frac{4}{3 {b}^{2}}$

Jun 15, 2017

$\frac{4}{3 {b}^{2}}$

#### Explanation:

$\frac{6 b}{4 {b}^{2}} \cdot \frac{8 {b}^{4}}{9 {b}^{5}}$

Let's simplify the expression by rewriting the expression and canceling out some terms:

$\frac{6}{4} \cdot \frac{b}{b} ^ 2 \cdot \frac{8}{9} \cdot {b}^{4} / {b}^{5} \implies {\cancel{6}}^{3} / {\cancel{4}}^{2} \cdot \frac{b}{b} ^ 2 \cdot \frac{8}{9} \cdot {b}^{4} / {b}^{5}$

$\frac{8}{9}$ can't be canceled so we will leave it as is. However, we can simplify the variable exponents by using one of the rules pertaining to exponents:

${a}^{-} n = \frac{1}{a} ^ n$

Using this, we can simplify the expression:

$\frac{3}{2} \cdot \frac{b}{b} ^ 2 \cdot \frac{8}{9} \cdot {b}^{4} / {b}^{5} \implies \frac{3}{2} \cdot {b}^{-} 1 \cdot \frac{8}{9} \cdot {b}^{-} 1 \implies \frac{3}{2} \cdot \frac{1}{b} \cdot \frac{8}{9} \cdot \frac{1}{b}$

We can then rearrange the expression like so:

$\frac{3}{2} \cdot \frac{1}{b} \cdot \frac{8}{9} \cdot \frac{1}{b} \implies \frac{3}{2} \cdot \frac{8}{9} \cdot \frac{1}{b} \cdot \frac{1}{b}$

We notice that we can cancel out the $3$ and the $9$ and the $2$ and the $8$ in the two terms at the front:

$\frac{3}{2} \cdot \frac{8}{9} \cdot \frac{1}{b} \cdot \frac{1}{b} \implies {\cancel{3}}^{1} / {\cancel{2}}^{1} \cdot {\cancel{8}}^{4} / {\cancel{9}}^{3} \cdot \frac{1}{b} \cdot \frac{1}{b}$

Then, we multiply the numerical terms and the variable terms together to obtain our answer:

$\frac{1}{1} \cdot \frac{4}{3} \cdot \frac{1}{b} \cdot \frac{1}{b} \implies \frac{4}{3} \cdot \frac{1}{b} ^ 2 \implies \frac{4}{3 {b}^{2}}$