How do you simplify #((3x^7)/(2y^12))^4# and write it using only positive exponents?

Question

Genesis Allen · Accepted Answer

See a solution process below:

Use, these rules of exponents to simplify the expression: #a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))# #((3x^7)/(2y^12))^4 => ((3^color(red)(1)x^color(red)(7))/(2^color(red)(1)y^color(red)(12)))^color(blue)(4) => (3^(color(red)(1)xxcolor(blue)(4))x^(color(red)(7)xxcolor(blue)(4)))/(2^(color(red)(1)xxcolor(blue)(4))y^(color(red)(12)xxcolor(blue)(4))) => (3^4x^28)/(2^4y^48) =># #(81x^28)/(16y^48)#

Avery Alexander · Answer

undefined To simplify ((3x^7)/(2y^12))^4 and write it using only positive exponents, raise each term inside the parentheses to the fourth power:

(3x^7)^4 = 3^4 * x^(7*4) = 81x^28
(2y^12)^4 = 2^4 * y^(12*4) = 16y^48

Then, divide the result of the first term by the result of the second term:

(81x^28) / (16y^48) = 81/16 * (x^28 / y^48)

So, the simplified expression with only positive exponents is:

(81/16) * (x^28 / y^48)

Eva Abramson · Answer

undefined To simplify $\left(\frac{3x^7}{2y^{12}}\right)^4$ and write it using only positive exponents, you apply the exponent rule for raising a fraction to a power:

$$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $$

Using this rule, we simplify the given expression as follows:

$$ \left(\frac{3x^7}{2y^{12}}\right)^4 = \frac{(3x^7)^4}{(2y^{12})^4} $$

Now, apply the exponent rule to the numerator and denominator separately:

$$ = \frac{3^4x^{7 \times 4}}{2^4y^{12 \times 4}} = \frac{81x^{28}}{16y^{48}} $$

Thus, the simplified form of $\left(\frac{3x^7}{2y^{12}}\right)^4$ with only positive exponents is $\frac{81x^{28}}{16y^{48}}$.