Simplifying Radicals Although we have been simplifying radicals
such as square roots, the process we use doesnt work as well for
higher roots like cube roots or fourth roots. The first examples
start with problems that could be simplified with other methods,
but we will introduce the new method which will be applied to more
complicated cases.
Slide 3
Example #1 Simplify each expression. A. Remember that square
roots have an index of 2. You can multiply roots with the same
index. 1.First break down all numbers into their prime
factorization. 2.Divide the remaining exponents of the factors by
the index. 3.If it divides evenly the factor can be pulled out
however many times it divides evenly, remainders remain under the
radical as the exponent of the same factor. For this problem, the
index of 2 divides into the exponent of 4 twice. That answer
becomes the exponent of the same factor outside the radical. The
exponent on the 3 is a 1 and does not divide by 2 evenly so it
remains under the radical.
Slide 4
Example #1 Simplify each expression. B. Even though they have
the same index they cannot be combined because they arent being
multiplied or divided. This time the index of 2 divided into the
exponent on the factor 3 once so the answer went on the factor of 3
outside the radical. The exponent on the 5 didnt divide evenly so
it remained under the radical. For the second radical, the index of
2 divided into the exponent of 3 once as well, so 1 factor of 5 was
pulled out, but the remainder was also a 1 so it became the
exponent on the 5 inside the radical. After being simplified, the
radicands are the same so they can be combined.
Slide 5
Example #1 Simplify each expression. C. The process for working
with variables is the same as with numbers. This time the index of
3 went into the exponents on the 2, 5, and x evenly, so 1 of each
factor was pulled out. The index divided into the exponent on the y
twice, so 2 became the exponent on the y outside the radical, but a
remainder of 1 was left and became the exponent on the y inside the
radical.
Slide 6
Example #1 Simplify each expression. D. Although FOIL could be
used, these are conjugates of each other and the middle terms would
cancel. The b > 0 is important because it tells us we dont have
to worry about imaginary values.
Slide 7
Example #1 Simplify each expression. E.
Slide 8
Example #1 Simplify each expression. F. The radicals in the
bottom are combined since the index is the same. Additionally, like
factors of 3 in the numerator and denominator can cancel for the
same reason. That is where the 3 2 comes from up top. Although a
square root of a negative number is imaginary, the cube root of 1
is 1.
Slide 9
Example #1 Simplify each expression. G. This problem can be
simplified using a slight number trick. Rewrite this number as a
product of 81 and a power of 10 similar to scientific notation. Be
careful when simplifying with negative exponents. If a remainder
remains, a negative goes on the exponent inside AND outside the
radical. (Note: This didnt apply to this problem)
Slide 10
Properties of Radical & Rational Exponent Expressions
Slide 11
Laws of Exponents
Slide 12
Example #2 Rewrite as a radical & simplify. Verify your
answer on a calculator. A. B. C.
Slide 13
Example #2 Rewrite as a radical & simplify. Verify your
answer on a calculator. D. E. F. Never put the negative from the
exponent in the index.
Slide 14
Example #3 Simplify each expression using only positive
exponents. A. B. Negative exponents need rewritten as positive by
moving them to the denominator. The c 1 in the denominator got
moved to the numerator and became c 1, then combined with c 4 to
become c 5. Notice when simplified, the negative exponent went
inside AND outside the radical.
Slide 15
Example #4 Simplify each expression using only positive
exponents. A. B. Rewrite numbers into their prime factorization.
Power to a power rule says to multiply exponents. Rewrite radicals
as fraction exponents. When multiplying like bases, ADD the
exponents.
Slide 16
Example #5 Simplify each expression using only positive
exponents. A. Distribute. Remember when multiplying like bases, ADD
the exponents.
Slide 17
Example #5 Simplify each expression using only positive
exponents. B. First groups with negative exponents are rearranged.
Groups with the same base multiplied together can be combined by
adding the exponents as well. The final answer has all like bases
combined with no negative exponents.
Slide 18
Example #6 Write the expression without radicals using only
positive exponents. A. Remember with power to a power to multiply
exponents. Add exponents for bases that are the same.
Slide 19
Example #6 Write the expression without radicals using only
positive exponents. B. Here we have two radicals buried inside each
other. The same rules apply, they become fractional exponents and
with power to a power you multiply the exponents together.
Slide 20
Example #6 Write the expression without radicals using only
positive exponents. C.
Slide 21
Example #7 Rationalize the denominator of each fraction.
(Remove the radical from the denominator) A. B. C. Multiply the top
and bottom by the conjugate of the denominator.
Slide 22
Example #8 Factor each expression. A. B. C. Factoring
expressions like this works very similarly to normal factoring.
Note: The exponent on each x term always matches that of the middle
term. Here the middle term is missing so this is a difference of
squares and they must add to be 0. This expression also factors
twice.
Slide 23
Example #9 Assume h 0, rationalize the numerator. Multiply the
top and bottom by the conjugate of the numerator this time.
