Calculus Examples

Evaluate the Limit limit as x approaches 0 of ((sin(x))/x)^(1/(x^2))
Step 1
Use the properties of logarithms to simplify the limit.
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Step 1.1
Rewrite as .
Step 1.2
Expand by moving outside the logarithm.
Step 2
Evaluate the limit.
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Step 2.1
Move the limit into the exponent.
Step 2.2
Combine and .
Step 3
Apply L'Hospital's rule.
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Step 3.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 3.1.1
Take the limit of the numerator and the limit of the denominator.
Step 3.1.2
Evaluate the limit of the numerator.
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Step 3.1.2.1
Move the limit inside the logarithm.
Step 3.1.2.2
Since and , apply the squeeze theorem.
Step 3.1.2.3
The natural logarithm of is .
Step 3.1.3
Evaluate the limit of the denominator.
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Step 3.1.3.1
Move the exponent from outside the limit using the Limits Power Rule.
Step 3.1.3.2
Evaluate the limit of by plugging in for .
Step 3.1.3.3
Raising to any positive power yields .
Step 3.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 3.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 3.2
Since is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Step 3.3
Find the derivative of the numerator and denominator.
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Step 3.3.1
Differentiate the numerator and denominator.
Step 3.3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.3.2.1
To apply the Chain Rule, set as .
Step 3.3.2.2
The derivative of with respect to is .
Step 3.3.2.3
Replace all occurrences of with .
Step 3.3.3
Multiply by the reciprocal of the fraction to divide by .
Step 3.3.4
Multiply by .
Step 3.3.5
Differentiate using the Quotient Rule which states that is where and .
Step 3.3.6
The derivative of with respect to is .
Step 3.3.7
Differentiate using the Power Rule which states that is where .
Step 3.3.8
Multiply by .
Step 3.3.9
Multiply by .
Step 3.3.10
Cancel the common factors.
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Step 3.3.10.1
Factor out of .
Step 3.3.10.2
Cancel the common factor.
Step 3.3.10.3
Rewrite the expression.
Step 3.3.11
Reorder terms.
Step 3.3.12
Differentiate using the Power Rule which states that is where .
Step 3.4
Multiply the numerator by the reciprocal of the denominator.
Step 3.5
Combine factors.
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Step 3.5.1
Multiply by .
Step 3.5.2
Raise to the power of .
Step 3.5.3
Raise to the power of .
Step 3.5.4
Use the power rule to combine exponents.
Step 3.5.5
Add and .
Step 4
Move the term outside of the limit because it is constant with respect to .
Step 5
Apply L'Hospital's rule.
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Step 5.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 5.1.1
Take the limit of the numerator and the limit of the denominator.
Step 5.1.2
Evaluate the limit of the numerator.
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Step 5.1.2.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 5.1.2.2
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 5.1.2.3
Move the limit inside the trig function because cosine is continuous.
Step 5.1.2.4
Move the limit inside the trig function because sine is continuous.
Step 5.1.2.5
Evaluate the limits by plugging in for all occurrences of .
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Step 5.1.2.5.1
Evaluate the limit of by plugging in for .
Step 5.1.2.5.2
Evaluate the limit of by plugging in for .
Step 5.1.2.5.3
Evaluate the limit of by plugging in for .
Step 5.1.2.6
Simplify the answer.
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Step 5.1.2.6.1
Simplify each term.
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Step 5.1.2.6.1.1
The exact value of is .
Step 5.1.2.6.1.2
Multiply by .
Step 5.1.2.6.1.3
The exact value of is .
Step 5.1.2.6.1.4
Multiply by .
Step 5.1.2.6.2
Add and .
Step 5.1.3
Evaluate the limit of the denominator.
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Step 5.1.3.1
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 5.1.3.2
Move the exponent from outside the limit using the Limits Power Rule.
Step 5.1.3.3
Move the limit inside the trig function because sine is continuous.
Step 5.1.3.4
Evaluate the limits by plugging in for all occurrences of .
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Step 5.1.3.4.1
Evaluate the limit of by plugging in for .
Step 5.1.3.4.2
Evaluate the limit of by plugging in for .
Step 5.1.3.5
Simplify the answer.
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Step 5.1.3.5.1
Raising to any positive power yields .
Step 5.1.3.5.2
The exact value of is .
Step 5.1.3.5.3
Multiply by .
Step 5.1.3.5.4
The expression contains a division by . The expression is undefined.
Undefined
Step 5.1.3.6
The expression contains a division by . The expression is undefined.
Undefined
Step 5.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 5.2
Since is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Step 5.3
Find the derivative of the numerator and denominator.
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Step 5.3.1
Differentiate the numerator and denominator.
Step 5.3.2
By the Sum Rule, the derivative of with respect to is .
Step 5.3.3
Evaluate .
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Step 5.3.3.1
Differentiate using the Product Rule which states that is where and .
Step 5.3.3.2
The derivative of with respect to is .
Step 5.3.3.3
Differentiate using the Power Rule which states that is where .
Step 5.3.3.4
Multiply by .
Step 5.3.4
Evaluate .
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Step 5.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.3.4.2
The derivative of with respect to is .
Step 5.3.5
Simplify.
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Step 5.3.5.1
Combine terms.
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Step 5.3.5.1.1
Subtract from .
Step 5.3.5.1.2
Add and .
Step 5.3.5.2
Reorder the factors of .
Step 5.3.6
Differentiate using the Product Rule which states that is where and .
Step 5.3.7
The derivative of with respect to is .
Step 5.3.8
Differentiate using the Power Rule which states that is where .
Step 5.3.9
Reorder terms.
Step 6
Apply L'Hospital's rule.
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Step 6.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 6.1.1
Take the limit of the numerator and the limit of the denominator.
Step 6.1.2
Evaluate the limit of the numerator.
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Step 6.1.2.1
Move the term outside of the limit because it is constant with respect to .
Step 6.1.2.2
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 6.1.2.3
Move the limit inside the trig function because sine is continuous.
Step 6.1.2.4
Evaluate the limits by plugging in for all occurrences of .
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Step 6.1.2.4.1
Evaluate the limit of by plugging in for .
Step 6.1.2.4.2
Evaluate the limit of by plugging in for .
Step 6.1.2.5
Simplify the answer.
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Step 6.1.2.5.1
The exact value of is .
Step 6.1.2.5.2
Multiply by .
Step 6.1.3
Evaluate the limit of the denominator.
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Step 6.1.3.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 6.1.3.2
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 6.1.3.3
Move the exponent from outside the limit using the Limits Power Rule.
Step 6.1.3.4
Move the limit inside the trig function because cosine is continuous.
Step 6.1.3.5
Move the term outside of the limit because it is constant with respect to .
Step 6.1.3.6
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 6.1.3.7
Move the limit inside the trig function because sine is continuous.
Step 6.1.3.8
Evaluate the limits by plugging in for all occurrences of .
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Step 6.1.3.8.1
Evaluate the limit of by plugging in for .
Step 6.1.3.8.2
Evaluate the limit of by plugging in for .
Step 6.1.3.8.3
Evaluate the limit of by plugging in for .
Step 6.1.3.8.4
Evaluate the limit of by plugging in for .
Step 6.1.3.9
Simplify the answer.
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Step 6.1.3.9.1
Simplify each term.
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Step 6.1.3.9.1.1
Raising to any positive power yields .
Step 6.1.3.9.1.2
The exact value of is .
Step 6.1.3.9.1.3
Multiply by .
Step 6.1.3.9.1.4
Multiply by .
Step 6.1.3.9.1.5
The exact value of is .
Step 6.1.3.9.1.6
Multiply by .
Step 6.1.3.9.2
Add and .
Step 6.1.3.9.3
The expression contains a division by . The expression is undefined.
Undefined
Step 6.1.3.10
The expression contains a division by . The expression is undefined.
Undefined
Step 6.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 6.2
Since is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Step 6.3
Find the derivative of the numerator and denominator.
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Step 6.3.1
Differentiate the numerator and denominator.
Step 6.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.3
Differentiate using the Product Rule which states that is where and .
Step 6.3.4
The derivative of with respect to is .
Step 6.3.5
Differentiate using the Power Rule which states that is where .
Step 6.3.6
Multiply by .
Step 6.3.7
Apply the distributive property.
Step 6.3.8
By the Sum Rule, the derivative of with respect to is .
Step 6.3.9
Evaluate .
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Step 6.3.9.1
Differentiate using the Product Rule which states that is where and .
Step 6.3.9.2
The derivative of with respect to is .
Step 6.3.9.3
Differentiate using the Power Rule which states that is where .
Step 6.3.10
Evaluate .
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Step 6.3.10.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.3.10.2
Differentiate using the Product Rule which states that is where and .
Step 6.3.10.3
The derivative of with respect to is .
Step 6.3.10.4
Differentiate using the Power Rule which states that is where .
Step 6.3.10.5
Multiply by .
Step 6.3.11
Simplify.
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Step 6.3.11.1
Apply the distributive property.
Step 6.3.11.2
Add and .
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Step 6.3.11.2.1
Move .
Step 6.3.11.2.2
Add and .
Step 6.3.11.3
Reorder terms.
Step 7
Apply L'Hospital's rule.
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Step 7.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 7.1.1
Take the limit of the numerator and the limit of the denominator.
Step 7.1.2
Evaluate the limit of the numerator.
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Step 7.1.2.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 7.1.2.2
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 7.1.2.3
Move the limit inside the trig function because cosine is continuous.
Step 7.1.2.4
Move the limit inside the trig function because sine is continuous.
Step 7.1.2.5
Evaluate the limits by plugging in for all occurrences of .
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Step 7.1.2.5.1
Evaluate the limit of by plugging in for .
Step 7.1.2.5.2
Evaluate the limit of by plugging in for .
Step 7.1.2.5.3
Evaluate the limit of by plugging in for .
Step 7.1.2.6
Simplify the answer.
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Step 7.1.2.6.1
Simplify each term.
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Step 7.1.2.6.1.1
The exact value of is .
Step 7.1.2.6.1.2
Multiply by .
Step 7.1.2.6.1.3
The exact value of is .
Step 7.1.2.6.1.4
Multiply by .
Step 7.1.2.6.2
Add and .
Step 7.1.3
Evaluate the limit of the denominator.
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Step 7.1.3.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 7.1.3.2
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 7.1.3.3
Move the exponent from outside the limit using the Limits Power Rule.
Step 7.1.3.4
Move the limit inside the trig function because sine is continuous.
Step 7.1.3.5
Move the term outside of the limit because it is constant with respect to .
Step 7.1.3.6
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 7.1.3.7
Move the limit inside the trig function because cosine is continuous.
Step 7.1.3.8
Move the term outside of the limit because it is constant with respect to .
Step 7.1.3.9
Move the limit inside the trig function because sine is continuous.
Step 7.1.3.10
Evaluate the limits by plugging in for all occurrences of .
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Step 7.1.3.10.1
Evaluate the limit of by plugging in for .
Step 7.1.3.10.2
Evaluate the limit of by plugging in for .
Step 7.1.3.10.3
Evaluate the limit of by plugging in for .
Step 7.1.3.10.4
Evaluate the limit of by plugging in for .
Step 7.1.3.10.5
Evaluate the limit of by plugging in for .
Step 7.1.3.11
Simplify the answer.
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Step 7.1.3.11.1
Simplify each term.
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Step 7.1.3.11.1.1
Raising to any positive power yields .
Step 7.1.3.11.1.2
Multiply by .
Step 7.1.3.11.1.3
The exact value of is .
Step 7.1.3.11.1.4
Multiply by .
Step 7.1.3.11.1.5
Multiply by .
Step 7.1.3.11.1.6
The exact value of is .
Step 7.1.3.11.1.7
Multiply by .
Step 7.1.3.11.1.8
The exact value of is .
Step 7.1.3.11.1.9
Multiply by .
Step 7.1.3.11.2
Add and .
Step 7.1.3.11.3
Add and .
Step 7.1.3.11.4
The expression contains a division by . The expression is undefined.
Undefined
Step 7.1.3.12
The expression contains a division by . The expression is undefined.
Undefined
Step 7.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 7.2
Since is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Step 7.3
Find the derivative of the numerator and denominator.
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Step 7.3.1
Differentiate the numerator and denominator.
Step 7.3.2
By the Sum Rule, the derivative of with respect to is .
Step 7.3.3
Evaluate .
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Step 7.3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.3.3.2
Differentiate using the Product Rule which states that is where and .
Step 7.3.3.3
The derivative of with respect to is .
Step 7.3.3.4
Differentiate using the Power Rule which states that is where .
Step 7.3.3.5
Multiply by .
Step 7.3.4
Evaluate .
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Step 7.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.3.4.2
The derivative of with respect to is .
Step 7.3.5
Simplify.
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Step 7.3.5.1
Apply the distributive property.
Step 7.3.5.2
Combine terms.
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Step 7.3.5.2.1
Multiply by .
Step 7.3.5.2.2
Multiply by .
Step 7.3.5.2.3
Subtract from .
Step 7.3.6
By the Sum Rule, the derivative of with respect to is .
Step 7.3.7
Evaluate .
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Step 7.3.7.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.3.7.2
Differentiate using the Product Rule which states that is where and .
Step 7.3.7.3
The derivative of with respect to is .
Step 7.3.7.4
Differentiate using the Power Rule which states that is where .
Step 7.3.8
Evaluate .
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Step 7.3.8.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.3.8.2
Differentiate using the Product Rule which states that is where and .
Step 7.3.8.3
The derivative of with respect to is .
Step 7.3.8.4
Differentiate using the Power Rule which states that is where .
Step 7.3.8.5
Multiply by .
Step 7.3.9
Evaluate .
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Step 7.3.9.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.3.9.2
The derivative of with respect to is .
Step 7.3.10
Simplify.
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Step 7.3.10.1
Apply the distributive property.
Step 7.3.10.2
Apply the distributive property.
Step 7.3.10.3
Combine terms.
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Step 7.3.10.3.1
Multiply by .
Step 7.3.10.3.2
Multiply by .
Step 7.3.10.3.3
Subtract from .
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Step 7.3.10.3.3.1
Move .
Step 7.3.10.3.3.2
Subtract from .
Step 7.3.10.3.4
Add and .
Step 8
Evaluate the limit.
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Step 8.1
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 8.2
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 8.3
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 8.4
Move the limit inside the trig function because sine is continuous.
Step 8.5
Move the term outside of the limit because it is constant with respect to .
Step 8.6
Move the limit inside the trig function because cosine is continuous.
Step 8.7
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 8.8
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 8.9
Move the exponent from outside the limit using the Limits Power Rule.
Step 8.10
Move the limit inside the trig function because cosine is continuous.
Step 8.11
Move the term outside of the limit because it is constant with respect to .
Step 8.12
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 8.13
Move the limit inside the trig function because sine is continuous.
Step 8.14
Move the term outside of the limit because it is constant with respect to .
Step 8.15
Move the limit inside the trig function because cosine is continuous.
Step 9
Evaluate the limits by plugging in for all occurrences of .
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Step 9.1
Evaluate the limit of by plugging in for .
Step 9.2
Evaluate the limit of by plugging in for .
Step 9.3
Evaluate the limit of by plugging in for .
Step 9.4
Evaluate the limit of by plugging in for .
Step 9.5
Evaluate the limit of by plugging in for .
Step 9.6
Evaluate the limit of by plugging in for .
Step 9.7
Evaluate the limit of by plugging in for .
Step 9.8
Evaluate the limit of by plugging in for .
Step 10
Simplify the answer.
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Step 10.1
Simplify the numerator.
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Step 10.1.1
The exact value of is .
Step 10.1.2
Multiply by .
Step 10.1.3
The exact value of is .
Step 10.1.4
Multiply by .
Step 10.1.5
Subtract from .
Step 10.2
Simplify the denominator.
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Step 10.2.1
Raising to any positive power yields .
Step 10.2.2
Multiply by .
Step 10.2.3
The exact value of is .
Step 10.2.4
Multiply by .
Step 10.2.5
Multiply by .
Step 10.2.6
The exact value of is .
Step 10.2.7
Multiply by .
Step 10.2.8
The exact value of is .
Step 10.2.9
Multiply by .
Step 10.2.10
Add and .
Step 10.2.11
Add and .
Step 10.3
Cancel the common factor of and .
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Step 10.3.1
Factor out of .
Step 10.3.2
Cancel the common factors.
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Step 10.3.2.1
Factor out of .
Step 10.3.2.2
Cancel the common factor.
Step 10.3.2.3
Rewrite the expression.
Step 10.4
Move the negative in front of the fraction.
Step 10.5
Multiply .
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Step 10.5.1
Multiply by .
Step 10.5.2
Multiply by .
Step 10.6
Rewrite the expression using the negative exponent rule .
Step 11
The result can be shown in multiple forms.
Exact Form:
Decimal Form: