Question

How do you multiply and simplify `\frac { a b ^ { - 4} \cdot 2b } { 3b a ^ { 2} }`?

Answer 1

`\frac{2}{3ab^4}`

Explanation:

Step 1 Identify like terms.
Exponentials with the same base are considered like terms.
`(\color(red)(a)\color(blue)(b)^{-4}\cdot\color(green)(2)\color(blue)(b))/(\color(green)(3)\color(blue)(b)\color(red)(a)^2)`

Step 2 Decide which exponent rules are relevant.

• Here are the rules associated with multiplication/division of same-base exponentials:
  `(x^a)/(x^b)=x^(a-b)` When dividing, SUBTRACT the exponent values.
  `(x^a)\cdot(x^b)=x^(a+b)` When multiplying, ADD the exponent values.
• Also... remember that when you have no visible exponent for a variable, the exponent is automatically 1 (`x=x^1`), and a negative exponent means "one over the positive version of the exponential" (`x^-2=1/(x^2)`).

Step 3 Apply relevant exponent rules.

• Now, we apply these rules to the problem. Let's group the like terms together, so it's easier to see.
  `\color(green)(2/3)\cdot(\color(red)(a))/(\color(red)(a)^2)\cdot((\color(blue)(b)^{-4})(\color(blue)(b)))/(\color(blue)(b))`
• Here, we see that variable `\color(red)(a)` will use the division rule, and variable `\color(blue)(b)` will use both multiplication and division rules.
  Disregarding the constants (the `\color(green)(2/3)`), we now have...
  `\frac(\color(red)(a))(\color(red)(a)^2)\rArr\color(red)(a)^{(1-2)}\rArr\color(red)(a)^-1\leftrightarrow\color(indianred)(1/a^1=1/a)`
  `((\color(blue)(b)^{-4})(\color(blue)(b)))/(\color(blue)(b))\rArr\frac{\color(blue)(b)^{(-4+1)}}{\color(blue)(b)}\rArr\frac{\color(blue)(b)^{-3}}{\color(blue)(b)}\rArr\color(blue)(b)^{(-3-1)}\rArr\color(blue)(b)^{-4}\leftrightarrow\color(steelblue)(1/b^4)`

Step 4 Substitute the simplified exponentials into the equation.

• We've simplified all the variable exponentials now, so we'll put those in place of the old ones.
  `\color(green)(2/3)\cdot\color(indianred)(1/a)\cdot\color(steelblue)(1/b^4)`
• Now, multiply the fractions you see, and you get...
  `\frac{\color(green)(2)\cdot\color(indianred)(1)\cdot\color(steelblue)(1)}{\color(green)(3)\cdot\color(indianred)(a)\cdot\color(steelblue)(b^4)}\rArr\frac{2}{3ab^4}`

Answer: `\frac{2}{3ab^4}`

Answer 2

The answer is `2/(3ab^4)`.

Explanation:

Simplify:

`(ab^(-4)*2b)/(3ba^2)=`

`(2ab^(-4)b)/(3ba^2)`

First use the product rule `a^ma^m=a^(m+n)`. If there is no exponent, the exponent is understood to be `1`.

`(2ab^((-4+1)))/(3ba^2)`

`(2ab^(-3))/(3ba^2)`

Next use the exponent quotient rule `a^m/a^n=a^(m-n)`. Remember, If there is no exponent, the exponent is understood to be `1`.

`(2a^(1-2)b^(-3-1))/3`

`(2a^(-1)b^(-4))/3`

Use negative exponent rule `a^(-m)=1/a^m`.

`2/(3ab^4)`