Question

How do you multiply `m^ { - 7/ 6} \cdot m ^ { 1/ 4}`?

Answer 1

`m^(-11/12)`

Explanation:

Using the `color(blue)"law of exponents"`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(a^mxxa^n=a^(m+n))color(white)(2/2)|)))`

`rArrm^(-7/6)xxm^(1/4)=m^(-7/6+1/4)`

`"Now " -7/6+1/4=-28/24+6/24=-22/24=-11/12`

`rArrm^(-7/6+1/4)=m^(-11/12)`

Answer 2

`1/m^(11/12)`

Explanation:

Recall the product rule for exponents:

`color(blue)(bar(ul(|color(white)(a/a)a^m*a^n=a^(m+n)color(white)(a/a)|)))`

When you are multiplying two powers with the same base, you add their exponent values together.

Applying the rule to the given question,

`m^(-7/6)*m^(1/4)`

The expression becomes `m` to the power of `-7/6+1/4`.

`=m^(-7/6+1/4)`

Since the fractions being added together do not have a common denominator, rewrite each fraction so that each one has the same denominator.

`=m^(-14/12+3/12)`

Evaluating,

`=m^(-11/12)`

However, expressions with negative exponents are usually simplified so that it only contains positive exponents.

Recall the negative exponent rule:

`color(blue)(bar(ul(|color(white)(a/a)a^-m=1/a^mcolor(white)(a/a)|)))`

Hence, `m^(-11/12)` becomes

`=color(green)( bar (ul ( | color(white)(a/a) color(black)(1/m^(11/12)) color(white)(a/a) | )))`