101B Simplifying Radicals

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10-1B Simplifying Radicals
In this section we are going to simplify rational
expressions with radicals in the denominator.
Algebra 1 by Gregory Hauca Glencoe
adapted from presentations by
Linda Stamper
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Simplifying Radicals
The simplest form of a radical expression is an
expression that has
No perfect square factors other than 1 in the
radicand.
No fractions in the radicand.
not simplified
No radicals in the denominator of a fraction.
not simplified
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2. Simplified radical expressions do not have
fractions under the radical (square root) sign.
Quotient Property of Radicals
Simplify the radical expression.
1. If possible, write the fraction in lowest
terms.
2. Use the quotient property to write the
numerator and denominator as square roots.
3. Simplify.
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Ex.1
Simplify the radical expression.
1. If possible, write the fraction in lowest
terms.
2. Use the quotient property to write the
numerator and denominator as square roots.
3. Simplify.
Ex.2
Simplify the radical expression.
1. If possible, write the fraction in lowest
terms.
2. Use the quotient property to write the
numerator and denominator as square roots.
3. Simplify.
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3. Simplified radical expressions do not have
radicals in the denominator.
Rationalizing the Denominator
This is a process used to eliminate a radical
from the denominator.
Key concept a radical multiplied by itself
results in a non-radical.
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Simplify the radical expression.
1. Rationalize the denominator by multiplying the
numerator and the denominator by the radical
shown in the denominator.
You cannot reduce numbers from inside and outside
the radical sign.
2. Multiply the numerators and multiply the
denominators.
3. Simplify.
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Simplify the radical expression.
Ex.1
1. Rationalize the denominator by multiplying the
numerator and the denominator by the radical
shown in the denominator.
Remember! You cannot reduce numbers from inside
and outside the radical sign.
2. Multiply the numerators and multiply the
denominators.
3. Simplify.
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Ex.2
Simplify the radical expression.
1. Rationalize the denominator by multiplying the
numerator and the denominator by the radical
shown in the denominator.
You cannot reduce numbers from inside and outside
the radical sign, so 2 and 10 cannot be reduced.
2. Multiply the numerators and multiply the
denominators.
You can reduce numbers that are outside the
radical sign.
3. Simplify.
Remember! You cannot reduce numbers from inside
and outside the radical sign.
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Ex.3
Simplify the radical expression.
1. If possible, write the fraction inside the
radical sign in lowest terms.
2. Write the numerator as a radical and write the
denominator as a radical.
3. Simplify.
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Ex.4
Simplify the radical expression.
1. If possible, write the fraction inside the
radical sign in lowest terms.
2. Write the numerator and denominator as a
radicals.
3. Rationalize the denominator by multiplying the
numerator and the denominator by the radical
shown in the denominator.
Remember! You cannot reduce numbers from inside
and outside the radical sign.
4. Multiply the numerators and multiply the
denominators.
5. Simplify.
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Ex.5
Simplify the radical expression.
1. If there is a perfect square in the
denominator do not reduce.
2. Write the numerator and denominator as a
radicals and simplify the perfect square
denominator.
Some questions will have more than one
simplifying steps.
3. Use the product property to simplify the
numerator.
Remember! You cannot reduce numbers from inside
and outside the radical sign.
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Ex.6
Simplify the radical expression.
1. Write the numerator and denominator as a
radicals and rationalize the denominator.
2. Simplify.
You can reduce. Both numbers are outside the
radical sign.
3. Factor the radicand using the greatest perfect
square factor.
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The Sum and Difference Pattern
(a b) (a b)
The SUM of a and b times the DIFFERENCE of a and
b.
a2 ab ab b2
After FOIL, the middle terms cancel because
theyre opposites.
The result is the difference of the squares of
the two original terms.
a2 b2
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To simplify expressions with radicals in the
denominator, you may be able to rewrite the
denominator as a rational number without changing
the value of the expression.
Multiply the denominator by its conjugate to
create a sum and difference pattern.
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Ex.7
Ex.8 Simplify.
1. Multiply by the conjugate of the denominator
to create a sum and difference pattern.
Do not distribute the numerator until the
denominator is simplified! You may be able to
reduce.
2. Multiply.
3. Use sum and difference pattern to multiply the
binomials.
4. Evaluate the powers and simplify.
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Ex.8
Ex.10 Simplify.
1. Multiply by the conjugate of the denominator
to create a sum and difference pattern.
2. Multiply.
3. Use sum and difference pattern to multiply the
binomials.
4. Evaluate the powers and simplify.
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Ex.9
Ex.9 Simplify.
1. Multiply by the conjugate of the denominator
to create a sum and difference pattern.
2. Multiply.
3. Use sum and difference pattern to multiply the
binomials.
4. Evaluate the powers and simplify.
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Ex.10
Ex.11 Simplify.
1. Multiply by the conjugate of the denominator
to create a sum and difference pattern.
2. Multiply.
3. Use sum and difference pattern to multiply the
binomials.
4. Evaluate the powers and simplify.
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Homework
HW10-A3 10-1 Skills Practice Handout 1-24.
(do on binder paper)
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To simplify expressions with radicals in the
denominator, you may be able to rewrite the
denominator as a rational number without changing
the value of the expression.
To eliminate the radical in the denominator we
rationalize the denominator
Multiply the denominator by its conjugate to
create a sum and difference pattern. (ab) is the
conjugate of (a-b)
The denominator is a binomial sum. Multiplying by
just the radical part will NOT clear the radical
denominator