Question

Perform the indicated operations. Assume all variables represent positive real numbers. ﻿81x6y34−16x10y34\sqrt[4]{$$81x^6y^3$} - \sqrt[4]{16x$$^{10}$$y^3$$}
481x6y3​−416x10y3​

Accepted Answer

Rewrite each term using fractional exponents to express the fourth roots: 
$$\sqrt[4]{81x$$^{6}$$y^{3}$$} = $$(81x^{6}y$$^{3}$$)^{\frac{1}$${4}}$$ and $$\sqrt[4]{$$16x^{10}y$$^{3}$$} = (16x$$^{10}$$y^{3}$$)^{\frac{1}$${4}}. $$Apply the exponent to each factor inside the parentheses separately using the property $$(abc)^m = a^m b^m c^m$$: 
$$(81)^{\frac{1}$${4}} $$(x^{6}$$)^{\frac{1}$${4}} (y$$^{3}$$)^{\frac{1}$${4}}$$ and (16$$)^{\frac{1}$${4}} (x$$^{10}$$)^{\frac{1}$${4}} $$(y^{3}$$)^{\frac{1}$${4}}. $$Simplify the numerical parts by finding the fourth root of 81 and 16: 
$$81 = 3^4$, so (81$$)^{\frac{1}$${4}} = 3$$, and $$16 = 2^4$, so (16$$)^{\frac{1}$${4}} = 2. $$Simplify the variable parts by multiplying the exponents: 
$$(x^{6}$$)^{\frac{1}$${4}} = x$$^{\frac{6}$${4}} = x$$^{\frac{3}$${2}}$$, 
$$(x^{10}$$)^{\frac{1}$${4}} = x$$^{\frac{10}$${4}} = x$$^{\frac{5}$${2}}$$, 
and $$(y^{3}$$)^{\frac{1}$${4}} = y$$^{\frac{3}$${4}}$$ for both terms. Rewrite the expression with the simplified parts and then perform the subtraction: 
$$3 x$$^{\frac{3}$${2}} y$$^{\frac{3}$${4}} - 2 x$$^{\frac{5}$${2}} y$$^{\frac{3}$${4}}$. 
From here, factor out the common terms if possible to simplify further.

Perform the indicated operations. Assume all variables represent positive real numbers. $\(\sqrt\)[4]{81x^6y^3} - \(\sqrt\)[4]{16x^{10}y^3}$

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주요 개념

Radical Expressions and Roots

Properties of Exponents

Operations with Like Radicals

Perform the indicated operations. Assume all variables represent positive real numbers. 81x6y34−16x10y34\(\sqrt\)[4]{81x^6y^3} - \(\sqrt\)[4]{16x^{10}y^3} 481x6y3−416x10y3