Use the product rule to multiply the following.7m24⋅2n4\sqrt[4]{7m^2}\cdot\sqrt[4]{2n}

14 m 2 n 4 \sqrt[4]{14m^2n} 4 14 m 2 n <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Use the product rule to multiply the following. 7 m 2 4 ⋅ 2 n 4 \sqrt[4]{7m^2}\cdot\sqrt[4]{2n}

we're asked to use the product rule to multiply the following. We have the fourth root of seven m squared times the fourth root of two n. So we're gonna go ahead and apply our product rule in order to condense the product of these two radicals into just one. So rewriting this here, this is going to be the fourth root because remember that's what both of these are. And then doing that multiplication underneath that one radical, this is seven m squared times two n. Doing some further simplification here, we can go ahead and multiply that seven times two to give us 14. So this becomes the fourth root of 14 m squared n, and that's our solution. Having used our product rule for nth roots to do that multiplication. Let us know if you have any questions.

Use the product rule to multiply the following. 7 m 2 4 ⋅ 2 n 4 \sqrt[4]{7m^2}\cdot\sqrt[4]{2n}

14 m 2 n 4 \sqrt[4]{14m^2n} 4 14 m 2 n <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

7 m 2 + 2 n 4 \sqrt[4]{7m^2+2n} 4 7 m 2 + 2 n <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

( 14 m 2 4 ) n \left(\sqrt[4]{14m^2}\right)n ( 4 14 m 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> ) n

7 m 2 4 + 2 n 4 \sqrt[4]{7m^2}+\sqrt[4]{2n} 4 7 m 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> + 4 2 n <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

9. Roots and Radicals

Simplifying Radical Expressions

Beginning Algebra

9. Roots and Radicals

Simplifying Radical Expressions

Struggling with Beginning Algebra?

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Multiple Choice

Use the product rule to multiply the following.
$\sqrt[4]{7 m^{2}} \cdot \sqrt[4]{2 n}$

$\sqrt[4]{7m^2+2n}$

$\sqrt[4]{14m^2n}$

$\left(\sqrt[4]{14m^2}\right)n$

$\sqrt[4]{7m^2}+\sqrt[4]{2n}$

Verified step by step guidance

Recognize that the problem involves multiplying two fourth roots: $\sqrt[4]{7m^2}$ and $\sqrt[4]{2n}$. The product rule for radicals states that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$ when the indices are the same.

Apply the product rule by multiplying the expressions inside the radicals: multiply \$7m^2$ by $2n$ to get $7m^2 \times 2n$.

Combine the multiplication inside the radical: \$7m^2 \times 2n = 14m^2n$.

Rewrite the product as a single fourth root: $\sqrt[4]{14m^2n}$.

Check if any simplification is possible by looking for perfect fourth powers inside the radical, but since none are obvious here, the expression $\sqrt[4]{14m^2n}$ is the simplified product.

5:28m

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