Exercises: Power Laws
- 1
Apply the power laws to simplify the following expressions:
For this task you need the following basic knowledge: Power Laws
↓ Use the power law with .
↓ Conclude the exponent.
For this task you need the following basic knowledge: Power Laws
↓ First apply the power laws to .
↓ Now apply the power law to .
↓ The term can be simplified even further by using the rule of the negative exponent, that is
For this task you need the following basic knowledge: Power Laws
↓ Apply the power law to .
↓ Write .
↓ Apply the power law to .
↓ Apply the power law .
For this task you need the following basic knowledge: Power Laws
↓ Apply the power law .
↓ Conclude the exponent
For this task you need the following basic knowledge: Power Laws
↓ Apply the power law to .
↓ Summarize the exponent of .
↓ Write .
↓ Use the power law .
↓ Conclude the base.
For this task you need the following basic knowledge: Power Laws
↓ Use the power law with base .
↓ Use the power law with base .
↓ Use the power law .
↓ Conclude the base.
- 2
Conclude as far as possible.
For this task you need the following basic knowledge: Powers
First Representation
↓ Shorten by cancelling three factors of .
Second Representation
↓ Apply the power laws.
For this task you need the following basic knowledge: Powers
Power laws
↓ Use the commutative law to group numbers and variables together.
↓ Apply the power laws.
For this task you need the following basic knowledge: Powers
↓ Write as .
↓ Apply the power laws.
For this task you need the following basic knowledge: Powers
↓ Write as .
↓ Apply the power laws.
Alternative solution
↓ Write as .
↓ Apply the power laws.
For this task you need the following basic knowledge: Powers
This term cannot be simplified further, since two different powers occur. However, what one could do is factorizing the term:
↓ Factorize .
For this task you need the following basic knowledge: Powers
First apply the power laws. You may get the minus out of the exponent by "flipping" the fractions like .
↓ Apply the power laws.
↓ Shorten by .
↓ Apply the power laws once more.
- 3
Find all terms that are equivalent to each other:
Term 1:
Term 2:
Term 3:
Term 4:
Term 5:
Term 6:
Term 7:
Term 8:
Term 9:
For this task you need the following basic knowledge: Power Laws
Term 1:
Term 2:
Term 3:
Apply the power laws.
Term 4:
Term 5:
The minus can be omitted here, because the power is even and .
Term 6:
Term 7:
Apply the power laws.
Term 8:
Apply the power laws.
Term 9:
Equivalent terms:
Term 1 and term 7 are both equal to .
Term 2 and term 8 are both equal to .
Term 3 and term 6 are both equal to .
- 4
Simplify the following terms.
For this task you need the following basic knowledge: Power Laws
↓ Apply the power laws step by step on the left three terms.
↓ For the sum, there is no power law. But you can conveniently express everything as a decimal number.
For this task you need the following basic knowledge: Power Laws
↓ Apply the power laws.
For this task you need the following basic knowledge: Power Laws
↓ Convert into fractions.
↓ Get both fractions to a common denominator.
For this task you need the following basic knowledge: Power Laws
↓ Apply the power laws.
↓ Get both fractions to a common denominator.
For this task you need the following basic knowledge: Power Laws
↓ Apply the power laws.
For this task you need the following basic knowledge: Power Laws
↓ Write as fractions.
↓ Multiply out.
↓ You can still shorten these fractions.
- 5
Simplify the following term using the power laws
For this task you need the following basic knowledge: Powers
↓ Conclude identical variables by applying the power laws.
↓ Conclude.
- 6
Simplify the following expressions with integer exponents as far as possible.
For this task you need the following basic knowledge: Powers
Write as .
↓ Apply the power laws.
↓ Factorize .
↓ Write as .
↓ Apply the power laws.
for
For this task you need the following basic knowledge: Powers
Apply the power laws.
↓ Write as , so you can shorten.
for
For this task you need the following basic knowledge: Powers
Factorize .
↓ Divide and apply the power laws.
↓ Multiply out.
↓ A negative exponent corresponds to a fraction.
for
For this task you need the following basic knowledge: Powers
Apply the power laws.
↓ Convert division into multiplication by "flipping" the fraction.
↓ Conclude.
Assuming that ,
For this task you need the following basic knowledge: Powers
Since is greater than , you can multiply by the reciprocal. For the value of
you then obtain
↓ Multiply out
↓ Factor out .
↓ Decompose the .
↓ Resolve the bracket and simplify.
↓ The factors of can be shortened.
Since is even, you get , so the final result reads , or equivalently, .
for
for
For this task you need the following basic knowledge: Powers
Resolve the brackets using the power laws.
↓ Write the division as a fraction.
↓ Resolve the double fraction.
↓ Shorten by using the power laws.
for
For this task you need the following basic knowledge: Powers
Expand the second fraction by to get a common denominator.
↓ Apply the power laws to .
↓ Get to a common denominator.
for
For this task you need the following basic knowledge: Powers
Apply the power laws.
↓ Resolve the bracket.
↓ Apply the power laws.
for
For this task you need the following basic knowledge: Powers
Get the fraction in the rightmost round bracket on a common denominator.
↓ Apply the power laws.
↓ Get the square bracket on a common denominator.
↓ Apply the power laws.
↓ Get the round bracket on a common denominator.
↓ Apply the power laws to conclude both brackets.
↓ Shorten by .
↓ Factor out a .
↓ Shorten.
Re-formulate the expression to a single fraction that does not contain any negative exponents.
For this task you need the following basic knowledge: Powers
Apply the power laws.
↓ Division can be done by "flipping the fraction".
↓ Resolve the double fraction.
↓ Finally, shorten.
- 7
Write as a decimal number.