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

2 Answers
Feb 22, 2017

\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}

Feb 22, 2017

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)