Rewrite <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Let <math><mstyle displaystyle="true"><mi>u</mi><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mstyle></math> . Substitute <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> for all occurrences of <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>6</mn><mi>u</mi><mo>-</mo><mn>16</mn></mstyle></math> using the AC method.

Consider the form <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> . Find a pair of integers whose product is <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> . In this case, whose product is <math><mstyle displaystyle="true"><mo>-</mo><mn>16</mn></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> .

Write the factored form using these integers.

Replace all occurrences of <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> with <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></mstyle></math> .

If any individual factor on the left side of the equation is equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the entire expression will be equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Set <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>-</mo><mn>2</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Solve <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to both sides of the equation.

Raise each side of the equation to the power of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> to eliminate the fractional exponent on the left side.

Simplify the exponent.

Simplify the left side.

Simplify <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></mstyle></math> .

Multiply the exponents in <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Simplify.

Simplify the right side.

Raise <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Set <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>8</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Solve <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> from both sides of the equation.

Raise each side of the equation to the power of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> to eliminate the fractional exponent on the left side.

Simplify the exponent.

Simplify the left side.

Simplify <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></mstyle></math> .

Multiply the exponents in <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Simplify.

Simplify the right side.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

The final solution is all the values that make <math><mstyle displaystyle="true"><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>-</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>8</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mstyle></math> true.

Exclude the solutions that do not make <math><mstyle displaystyle="true"><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mn>6</mn><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>-</mo><mn>16</mn><mo>=</mo><mn>0</mn></mstyle></math> true.

Do you know how to Evaluate n^(1/2)+6n^(1/4)-16=0? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.