Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn></mstyle></math> contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part <math><mstyle displaystyle="true"><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mstyle></math> then find LCM for the variable part <math><mstyle displaystyle="true"><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msup></mstyle></math> .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Since <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> has no factors besides <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The number <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is not a prime number because it only has one positive factor, which is itself.

Not prime

Since <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> has no factors besides <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The LCM of <math><mstyle displaystyle="true"><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mstyle></math> is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factors for <math><mstyle displaystyle="true"><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> are <math><mstyle displaystyle="true"><mi>m</mi><mo>⋅</mo><mi>m</mi></mstyle></math> , which is <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> multiplied by each other <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> times.

The factor for <math><mstyle displaystyle="true"><msup><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msup></mstyle></math> is <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> itself.

The LCM of <math><mstyle displaystyle="true"><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msup></mstyle></math> is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> .

The LCM for <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn></mstyle></math> is the numeric part <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> multiplied by the variable part.

Multiply each term in <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> in order to remove all the denominators from the equation.

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>⋅</mo><mrow><mo>(</mo><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mstyle></math> .

Rewrite using the commutative property of multiplication.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of <math><mstyle displaystyle="true"><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Simplify each term.

Rewrite using the commutative property of multiplication.

Combine <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> out of <math><mstyle displaystyle="true"><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Move the leading negative in <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> into the numerator.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite the equation as <math><mstyle displaystyle="true"><mn>2</mn><mi>m</mi><mo>-</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn></mstyle></math> .

Move <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> to the left side of the equation by subtracting it from both sides.

Factor the left side of the equation.

Let <math><mstyle displaystyle="true"><mi>u</mi><mo>=</mo><mi>m</mi></mstyle></math> . Substitute <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> for all occurrences of <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> .

Factor by grouping.

Reorder terms.

For a polynomial of the form <math><mstyle displaystyle="true"><mi>a</mi><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> , rewrite the middle term as a sum of two terms whose product is <math><mstyle displaystyle="true"><mi>a</mi><mo>⋅</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>⋅</mo><mo>-</mo><mn>1</mn><mo>=</mo><mn>1</mn></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>2</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mi>u</mi></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> as <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> plus <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, <math><mstyle displaystyle="true"><mo>-</mo><mi>u</mi><mo>+</mo><mn>1</mn></mstyle></math> .

Replace all occurrences of <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> with <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> .

Replace the left side with the factored expression.

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 the first factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solve.

Set the first factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from both sides of the equation.

Multiply each term in <math><mstyle displaystyle="true"><mo>-</mo><mi>m</mi><mo>=</mo><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math>

Multiply each term in <math><mstyle displaystyle="true"><mo>-</mo><mi>m</mi><mo>=</mo><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mi>m</mi><mo>)</mo></mrow><mo>⋅</mo><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

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

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