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Simplify the quotient of powers $\frac{\sqrt{2x^{12}y^{21}z^8}}{40\sqrt{xyz}}$

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Solution

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Final answer to the problem

$\frac{7}{198}\sqrt{z^{7}}\left(\frac{11}{2}\sqrt{x^{9}}y^{10}+10\sqrt{x^{11}}y^{9}\right)$
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Step-by-step Solution

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The power of a product is equal to the product of it's factors raised to the same power

$\frac{d}{dx}\left(\frac{\sqrt{2}x^{6}\sqrt{y^{21}}z^{4}}{40\sqrt{x}\sqrt{y}\sqrt{z}}\right)$

Learn how to solve differential calculus problems step by step online.

$\frac{d}{dx}\left(\frac{\sqrt{2}x^{6}\sqrt{y^{21}}z^{4}}{40\sqrt{x}\sqrt{y}\sqrt{z}}\right)$

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Learn how to solve differential calculus problems step by step online. Simplify the quotient of powers ((2x^12y^21z^8)^1/2)/(40(xyz)^1/2). The power of a product is equal to the product of it's factors raised to the same power. Simplify the fraction \frac{\sqrt{2}x^{6}\sqrt{y^{21}}z^{4}}{40\sqrt{x}\sqrt{y}\sqrt{z}} by x. Simplify the fraction \frac{\sqrt{2}\sqrt{x^{11}}\sqrt{y^{21}}z^{4}}{40\sqrt{y}\sqrt{z}} by y. Simplify the fraction \frac{\sqrt{2}\sqrt{x^{11}}y^{10}z^{4}}{40\sqrt{z}} by z.

Final answer to the problem

$\frac{7}{198}\sqrt{z^{7}}\left(\frac{11}{2}\sqrt{x^{9}}y^{10}+10\sqrt{x^{11}}y^{9}\right)$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Find the derivativeFind derivative of (2x^12^0.5)/40xy^0.5 using the product ruleFind derivative of (2x^12^0.5)/40xy^0.5 using the quotient ruleFind derivative of (2x^12^0.5)/40xy^0.5 using logarithmic differentiationFind derivative of (2x^12^0.5)/40xy^0.5 using the definition

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Plotting: $\frac{7}{198}\sqrt{z^{7}}\left(\frac{11}{2}\sqrt{x^{9}}y^{10}+10\sqrt{x^{11}}y^{9}\right)$

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Similar Problems Solved

$\frac{\sqrt{2x^{12} y^{21} z^8}}{40\sqrt{xy\cdot z}}$

$\frac{\left(x^{\frac{1}{5}} y^{\frac{3}{2}}\right)^{10}}{\left(y^{\left(-1\right)\cdot\frac{2}{5}} x^{\frac{2}{3}}\right)^5}$

$\frac{\left(4x^{121\left(-1\right)} y^4 z^8\right)^3}{\left(9x y^3 z^5\right)^4}$

$\frac{\left(-5x^3 y^2\right)^3\left(x y^5\right)^2}{\left(-5x^{-1}\cdot y\right)^{-1}}$

$\frac{\left(x^{\left(-1\right)\cdot\frac{1}{5}} y^{\frac{3}{2}}\right)^{10}}{\left(y^{\left(-1\right)\cdot\frac{2}{5}} x^{\frac{2}{3}}\right)^5}$

$\frac{\left(4x^3\right)^2}{\left(121\left(-1\right) x^5\right)^3}$

$\frac{24\left(x+y\right)^3\left(x-y\right)^2}{30\left(x+y\right)^2\left(x-y\right)^3}$

Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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