Question

$\huge\text{Please help!}$
Simplify:
$\bold{(x^{2} +8xy+16y^{2})^{\frac{1}{3} } (x+4y)}^{\frac{1}{3} }$
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$\bold{M^ag^ic^al\:^}$

Answer 1

Answer:

x + 4y

Step-by-step explanation:

Hey there! First, we have to recall back laws of exponent.

$\displaystyle \large{a^{\frac{m}{n}} = \sqrt[n]{a^m} }$

Now simplify the expressions in surds form.

$\displaystyle \large{\sqrt[3]{x^2+8xy+16y^2} \cdot \sqrt[3]{x+4y} }$

From x²+8xy+16y², we can convert the expression to perfect square.

Perfect Square

$\displaystyle \large{x^2+2xb+b^2 = (x+b)^2}$

Therefore, from the expression.

$\displaystyle \large{x^2+8xy+16y^2 = (x+4y)^2}$

Thus:

$\displaystyle \large{\sqrt[3]{(x+4y)^2} \cdot \sqrt[3]{x+4y}$

Because both have same surds, multiply them in one.

Surd Property I

$\displaystyle \large{\sqrt[n]{x} \cdot \sqrt[n]{y} =\sqrt[n]{xy} }$

Therefore:

$\displaystyle \large{\sqrt[3]{(x+4y)^2(x+4y)}$

Since both are like-terms and multiplying each other, we can apply one of exponent laws.

Exponent Laws II

$\displaystyle \large{a^m \cdot a^n = a^{m+n}}$

Therefore, we have $\displaystyle \large{\sqrt[3]{(x+4y)^{2+1} } \to \sqrt[3]{(x+4y)^3} }$

Simplify the expression, we have cube expression inside the cube root. Therefore, we simplify as x+4y as we cancel cube and cube root.

Let me know if you have any questions through comments!

Answer 2

$\bold{x + 4y}$

Explanation :-

${(x^{2} +8xy+16y^{2})^{\frac{1}{3} } (x+4y)}^{\frac{1}{3} }$

Rewriting it into the expression of $a^{2} + 2ab + {b}^{2}$,

$x^{2} + 2x \times 4y + (4y)^{2})^{\frac{1}{3} }(x + 4y)^{ \frac{1}{3} }$

$= > ((x + 4y)^{2})^{ \frac{1}{3} }(x + 4y)^{ \frac{1}{3} }$

Now multiplying the fractions as,

$= > ((x + 4y))^{ \frac{2}{3} }(x + 4y)^{ \frac{1}{3} }$

Now as the bases of the fractions are same, we can multiply directly,

$= > (x + 4y)^{ \frac{2}{3} + \frac{1}{3} }$

$= > (x + 4y)^{ \frac{3}{3} }$

As when same numerator and denominator of a fraction is 1, so,

$= > (x + 4y)^1$

As 1 can be neglected (since it has no specific need in the equation), so,

$= > x + 4y$

Hence, the answer is $\bold{x + 4y}$