Given x > 0 and y > 0, select the expression that is equivalent to root(4)(256x^(2)y^(8)) 64x^(2)y^((1)/(2)) 4x^(2)y^((1)/(2)) 64x^((1)/(2))y^(2) 4x^((1)/(2))y^(2)

Write Expression with Radical Notation: Write the given expression with radical notation. The fourth root of 256x^2y^8 can be written as (256x^2y^8)^(1/4).Simplify Constant Term: Simplify the constant term under the fourth root. 256 is a perfect fourth power since 256 = 4^4. Therefore, the fourth root of 256 is 4. (256x^2y^8)^(1/4) = (4^4x^2y^8)^(1/4)Apply Exponent Property: Apply the property of exponents (a^m)^n = a^(m*n) to simplify the expression. (4^4x^2y^8)^(1/4) = 4^(4*(1/4)) * x^(2*(1/4)) * y^(8*(1/4))Simplify Exponents: Simplify the exponents by multiplying them. 4^(4*(1/4)) = 4^1 = 4 x^(2*(1/4)) = x^(1/2) y^(8*(1/4)) = y^2Combine Simplified Terms: Combine the simplified terms. 4 * x^(1/2) * y^2Compare with Given Options: Compare the result with the given options. The expression 4 * x^(1/2) * y^2 matches the option ＂4x^((1)/(2))y^(2)＂.

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Given
x > 0 and
y > 0, select the expression that is equivalent to

root(4)(256x^(2)y^(8))

64x^(2)y^((1)/(2))

4x^(2)y^((1)/(2))

64x^((1)/(2))y^(2)

4x^((1)/(2))y^(2)

Given x>0 and y>0 , select the expression that is equivalent to $\newline$ $\sqrt[4]{256 x^{2} y^{8}}$ $\newline$ $64 x^{2} y^{\frac{1}{2}}$ $\newline$ $4 x^{2} y^{\frac{1}{2}}$ $\newline$ $64 x^{\frac{1}{2}} y^{2}$ $\newline$ $4 x^{\frac{1}{2}} y^{2}$

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Math Problems

Algebra 2

Multiplication with rational exponents

Full solution

Q. Given

x>0

and

y>0

, select the expression that is equivalent to

\newline

\sqrt[4]{256 x^{2} y^{8}}

\newline

64 x^{2} y^{\frac{1}{2}}

\newline

4 x^{2} y^{\frac{1}{2}}

\newline

64 x^{\frac{1}{2}} y^{2}

\newline

4 x^{\frac{1}{2}} y^{2}

Write Expression with Radical Notation: Write the given expression with radical notation. $\newline$ The fourth root of $256x^2y^8$ can be written as $(256x^2y^8)^{\frac{1}{4}}$ .
Simplify Constant Term: Simplify the constant term under the fourth root. $\newline$ $256$ is a perfect fourth power since $256 = 4^4$ . Therefore, the fourth root of $256$ is $4$ . $\newline$ $(256x^2y^8)^{1/4} = (4^4x^2y^8)^{1/4}$
Apply Exponent Property: Apply the property of exponents $(a^m)^n = a^{m*n}$ to simplify the expression. $\newline$ $(4^{4}x^{2}y^{8})^{\frac{1}{4}} = 4^{4*(\frac{1}{4})} * x^{2*(\frac{1}{4})} * y^{8*(\frac{1}{4})}$
Simplify Exponents: Simplify the exponents by multiplying them. $\newline$ $4^{4\times(\frac{1}{4})} = 4^1 = 4$ $\newline$ $x^{2\times(\frac{1}{4})} = x^{\frac{1}{2}}$ $\newline$ $y^{8\times(\frac{1}{4})} = y^2$
Combine Simplified Terms: Combine the simplified terms. $4 \times x^{\frac{1}{2}} \times y^2$
Compare with Given Options: Compare the result with the given options. $\newline$ The expression $4 \times x^{\frac{1}{2}} \times y^2$ matches the option ＂ $4x^{\left(\frac{1}{2}\right)}y^{2}$ ＂.