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Sagot :
Answer:
x + 4y
Step-by-step explanation:
Hey there! First, we have to recall back laws of exponent.
[tex]\displaystyle \large{a^{\frac{m}{n}} = \sqrt[n]{a^m} }[/tex]
Now simplify the expressions in surds form.
[tex]\displaystyle \large{\sqrt[3]{x^2+8xy+16y^2} \cdot \sqrt[3]{x+4y} }[/tex]
From x²+8xy+16y², we can convert the expression to perfect square.
Perfect Square
[tex]\displaystyle \large{x^2+2xb+b^2 = (x+b)^2}[/tex]
Therefore, from the expression.
[tex]\displaystyle \large{x^2+8xy+16y^2 = (x+4y)^2}[/tex]
Thus:
[tex]\displaystyle \large{\sqrt[3]{(x+4y)^2} \cdot \sqrt[3]{x+4y}[/tex]
Because both have same surds, multiply them in one.
Surd Property I
[tex]\displaystyle \large{\sqrt[n]{x} \cdot \sqrt[n]{y} =\sqrt[n]{xy} }[/tex]
Therefore:
[tex]\displaystyle \large{\sqrt[3]{(x+4y)^2(x+4y)}[/tex]
Since both are like-terms and multiplying each other, we can apply one of exponent laws.
Exponent Laws II
[tex]\displaystyle \large{a^m \cdot a^n = a^{m+n}}[/tex]
Therefore, we have [tex]\displaystyle \large{\sqrt[3]{(x+4y)^{2+1} } \to \sqrt[3]{(x+4y)^3} }[/tex]
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!
[tex]\bold{x + 4y}[/tex]
Explanation :-
[tex]{(x^{2} +8xy+16y^{2})^{\frac{1}{3} } (x+4y)}^{\frac{1}{3} }[/tex]
Rewriting it into the expression of [tex]a^{2} + 2ab + {b}^{2} [/tex],
[tex]x^{2} + 2x \times 4y + (4y)^{2})^{\frac{1}{3} }(x + 4y)^{ \frac{1}{3} }[/tex]
[tex] = > ((x + 4y)^{2})^{ \frac{1}{3} }(x + 4y)^{ \frac{1}{3} }[/tex]
Now multiplying the fractions as,
[tex] = > ((x + 4y))^{ \frac{2}{3} }(x + 4y)^{ \frac{1}{3} }[/tex]
Now as the bases of the fractions are same, we can multiply directly,
[tex] = > (x + 4y)^{ \frac{2}{3} + \frac{1}{3} }[/tex]
[tex] = > (x + 4y)^{ \frac{3}{3} }[/tex]
As when same numerator and denominator of a fraction is 1, so,
[tex] = > (x + 4y)^1[/tex]
As 1 can be neglected (since it has no specific need in the equation), so,
[tex] = > x + 4y[/tex]
Hence, the answer is [tex]\bold{x + 4y}[/tex]
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