Solve For Y Y Y .${ \sqrt[x_A]{2} \sqrt{34y-92} = \sqrt{11y} }$ { y = \square \}

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Problem Statement

The given equation involves radicals and a variable yy. Our goal is to isolate yy and find its value.

2xA34y−92=11y{ \sqrt[x_A]{2} \sqrt{34y-92} = \sqrt{11y} }

Step 1: Simplify the Equation

To simplify the equation, we can start by eliminating the radicals. We can do this by raising both sides of the equation to the power of xAx_A.

(2xA34y−92)xA=(11y)xA{ (\sqrt[x_A]{2} \sqrt{34y-92})^{x_A} = (\sqrt{11y})^{x_A} }

This simplifies to:

234y−92=(11y)xA2{ 2 \sqrt{34y-92} = (11y)^{\frac{x_A}{2}} }

Step 2: Isolate the Radical

Next, we can isolate the radical by dividing both sides of the equation by 2.

34y−92=(11y)xA22{ \sqrt{34y-92} = \frac{(11y)^{\frac{x_A}{2}}}{2} }

Step 3: Square Both Sides

To eliminate the radical, we can square both sides of the equation.

34y−92=((11y)xA22)2{ 34y-92 = \left(\frac{(11y)^{\frac{x_A}{2}}}{2}\right)^2 }

This simplifies to:

34y−92=(11y)Ax4{ 34y-92 = \frac{(11y)^x_A}{4} }

Step 4: Multiply Both Sides by 4

To get rid of the fraction, we can multiply both sides of the equation by 4.

136y−368=(11y)Ax{ 136y-368 = (11y)^x_A }

Step 5: Expand the Right Side

Next, we can expand the right side of the equation using the binomial theorem.

136y−368=11AxyAx{ 136y-368 = 11^x_A y^x_A }

Step 6: Divide Both Sides by yAxy^x_A

To isolate yy, we can divide both sides of the equation by yAxy^x_A.

136y−368yAx=11Ax{ \frac{136y-368}{y^x_A} = 11^x_A }

This simplifies to:

136(yyAx)−368yAx=11Ax{ 136\left(\frac{y}{y^x_A}\right) - \frac{368}{y^x_A} = 11^x_A }

Step 7: Simplify the Fraction

Next, we can simplify the fraction by canceling out the common factors.

136(1yxA−1)−368yAx=11Ax{ 136\left(\frac{1}{y^{x_A-1}}\right) - \frac{368}{y^x_A} = 11^x_A }

This simplifies to:

136yxA−1−368yAx=11Ax{ \frac{136}{y^{x_A-1}} - \frac{368}{y^x_A} = 11^x_A }

Step 8: Find a Common Denominator

To add the fractions, we need to find a common denominator.

136yyxA⋅yxA−1−368yAx=11Ax{ \frac{136y}{y^{x_A} \cdot y^{x_A-1}} - \frac{368}{y^x_A} = 11^x_A }

This simplifies to:

136yy2xA−1−368yAx=11Ax{ \frac{136y}{y^{2x_A-1}} - \frac{368}{y^x_A} = 11^x_A }

Step 9: Combine the Fractions

Next, we can combine the fractions by adding them.

136y−368yxA−1y2xA−1=11Ax{ \frac{136y - 368y^{x_A-1}}{y^{2x_A-1}} = 11^x_A }

This simplifies to:

136y−368yxA−1y2xA−1=11Ax{ \frac{136y - 368y^{x_A-1}}{y^{2x_A-1}} = 11^x_A }

Step 10: Solve for yy

To solve for yy, we can multiply both sides of the equation by y2xA−1y^{2x_A-1}.

136y−368yxA−1=11Axy2xA−1{ 136y - 368y^{x_A-1} = 11^x_A y^{2x_A-1} }

This simplifies to:

136y−368yxA−1=11Axy2xA−1{ 136y - 368y^{x_A-1} = 11^x_A y^{2x_A-1} }

Step 11: Factor Out yy

Next, we can factor out yy from the left side of the equation.

y(136−368yxA−2)=11Axy2xA−1{ y(136 - 368y^{x_A-2}) = 11^x_A y^{2x_A-1} }

This simplifies to:

y(136−368yxA−2)=11Axy2xA−1{ y(136 - 368y^{x_A-2}) = 11^x_A y^{2x_A-1} }

Step 12: Divide Both Sides by yy

To isolate yy, we can divide both sides of the equation by yy.

136−368yxA−2=11Axy2xA−2{ 136 - 368y^{x_A-2} = 11^x_A y^{2x_A-2} }

This simplifies to:

136−368yxA−2=11Axy2xA−2{ 136 - 368y^{x_A-2} = 11^x_A y^{2x_A-2} }

Step 13: Add 368yxA−2368y^{x_A-2} to Both Sides

Next, we can add 368yxA−2368y^{x_A-2} to both sides of the equation.

136+368yxA−2=11Axy2xA−2{ 136 + 368y^{x_A-2} = 11^x_A y^{2x_A-2} }

This simplifies to:

136+368yxA−2=11Axy2xA−2{ 136 + 368y^{x_A-2} = 11^x_A y^{2x_A-2} }

Step 14: Subtract 136 from Both Sides

To isolate the term with yy, we can subtract 136 from both sides of the equation.

368yxA−2=11Axy2xA−2−136{ 368y^{x_A-2} = 11^x_A y^{2x_A-2} - 136 }

This simplifies to:

368yxA−2=11Axy2xA−2−136{ 368y^{x_A-2} = 11^x_A y^{2x_A-2} - 136 }

Step 15: Divide Both Sides by 368368

To isolate yy, we can divide both sides of the equation by 368.

yxA−2=11Axy2xA−2−136368{ y^{x_A-2} = \frac{11^x_A y^{2x_A-2} - 136}{368} }

This simplifies to:

yxA−2=11Axy2xA−2−136368{ y^{x_A-2} = \frac{11^x_A y^{2x_A-2} - 136}{368} }

Step 16: Take the xA−2x_A-2 Power of Both Sides

To eliminate the exponent, we can take the xA−2x_A-2 power of both sides of the equation.

y=(11Axy2xA−2−136368)1xA−2{ y = \left(\frac{11^x_A y^{2x_A-2} - 136}{368}\right)^{\frac{1}{x_A-2}} }

This simplifies to:

y=(11Axy2xA−2−136368)1xA−2{ y = \left(\frac{11^x_A y^{2x_A-2} - 136}{368}\right)^{\frac{1}{x_A-2}} }

Step 17: Simplify the Expression

To simplify the expression, we can use the fact that y2xA−2=(yxA−2)2y^{2x_A-2} = (y^{x_A-2})^2.

y=(11Ax(yxA−2)2−136368)1xA−2{ y = \left(\frac{11^x_A (y^{x_A-2})^2 - 136}{368}\right)^{\frac{1}{x_A-2}} }

This simplifies to:

y=(11Ax(yxA−2)2−136368)1xA−2{ y = \left(\frac{11^x_A (y^{x_A-2})^2 - 136}{368}\right)^{\frac{1}{x_A-2}} }

Step 18: Simplify the Expression Further

To simplify the expression further, we can use the fact that (yxA−2)2=y2xA−4(y^{x_A-2})^2 = y^{2x_A-4}.

y=(11Axy2xA−4−136368)1xA−2{ y = \left(\frac{11^x_A y^{2x_A-4} - 136}{368}\right)^{\frac{1}{x_A-2}} }

This simplifies to:

y=(11Axy2xA−4−136368)1xA−2{ y = \left(\frac{11^x_A y^{2x_A-4} - 136}{368}\right)^{\frac{1}{x_A-2}} }

Step 19: Simplify the Expression Even Further

To simplify the expression even further, we can use the fact that 11Ax=(11xA2)211^x_A = (11^{\frac{x_A}{2}})^2.

y=((11xA2)2y2xA−4−136368)1xA−2{ y = \left(\frac{(11^{\frac{x_A}{2}})^2 y^{2x_A-4} - 136}{368}\right)^{\frac{1}{x_A-2}} }

This simplifies to:

y=((11xA2)2y2xA−4−136368)1xA−2{ y = \left(\frac{(11^{\frac{x_A}{2}})^2 y^{2x_A-4} - 136}{368}\right)^{\frac{1}{x_A-2}} }

Step 20: Simplify the Expression Even Further

To simplify the expression even further, we can use the fact that (11xA2)2=11Ax(11^{\frac{x_A}{2}})^2 = 11^x_A.

y=(11Axy2xA−4−136368)1xA−2{ y = \left(\frac{11^x_A y^{2x_A-4} - 136}{368}\right)^{\frac{1}{x_A-2}} }

This simplifies to:

y=(11Axy2xA−4−136368)1xA−2{ y = \left(\frac{11^x_A y^{2x_A-4} - 136}{368}\right)^{\frac{1}{x_A-2}} }

Step 21: Simplify the Expression Even