How do you multiply and simplify \frac { a b ^ { - 4} \cdot 2b } { 3b a ^ { 2} }?
2 Answers
\frac{2}{3ab^4}
Explanation:
Step 1 Identify like terms.
Exponentials with the same base are considered like terms.
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}
The answer is
Explanation:
Simplify:
First use the product rule
Next use the exponent quotient rule
Use negative exponent rule