Whole root of (√3−1)^(2)−(√3−1)^(2)+4^(2)
Answers
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Answer:
Step-by-step explanation:
Example 1: Multiply:
√
2
⋅
√
6
.
Solution: This problem is a product of two square roots. Apply the product rule for radicals and then simplify.
Answer:
2
√
3
Example 2: Multiply:
3
√
9
⋅
3
√
6
.
Solution: This problem is a product of cube roots. Apply the product rule for radicals and then simplify.
Answer:
3
3
√
2
Often there will be coefficients in front of the radicals.
Example 3: Multiply:
2
√
3
⋅
5
√
2
.
Solution: Using the product rule for radicals and the fact that multiplication is commutative, we can multiply the coefficients and the radicands as follows.
Typically, the first step involving the application of the commutative property is not shown.
Answer:
10
√
6
Example 4: Multiply:
−
2
3
√
5
x
⋅
3
3
√
25
x
2
.
Solution:
Answer:
−
30
x
Use the distributive property when multiplying rational expressions with more than one term.
Example 5: Multiply:
4
√
3
(
2
√
3
−
3
√
6
)
.
Solution: Apply the distributive property and multiply each term by
4
√
3
.
Answer:
24
−
36
√
2
Example 6: Multiply:
3
√
4
x
2
(
3
√
2
x
−
5
3
√
4
x
2
)
.
Solution: Apply the distributive property and then simplify the result.
Answer:
2
x
−
10
x
⋅
3
√
2
x
The process for multiplying radical expressions with multiple terms is the same process used when multiplying polynomials. Apply the distributive property, simplify each radical, and then combine like terms.
Example 7: Multiply:
(
√
5
+
2
)
(
√
5
−
4
)
.
Solution: Begin by applying the distributive property.
Answer:
−
3
−
2
√
5
Example 8: Multiply:
(
3
√
x
−
√
y
)
2
.
Solution:
Answer:
9
x
−
6
√
x
y
+
y
Try this! Multiply:
(
2
√
3
+
5
√
2
)
(
√
3
−
2
√
6
)
.
Answer:
6
−
12
√
2
+
5
√
6
−
20
√
3
Video Solution
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The expressions
(
a
+
b
)
and
(
a
−
b
)
are called conjugates. When multiplying conjugates, the sum of the products of the inner and outer terms results in 0.
Example 9: Multiply:
(
√
2
+
√
5
)
(
√
2
−
√
5
)
.
Solution: Apply the distributive property and then combine like terms.
Answer: −3
It is important to note that when multiplying conjugate radical expressions, we obtain a rational expression. This is true in general and is often used in our study of algebra.
Therefore, for nonnegative real numbers a and b, we have the following property:
Dividing Radical Expressions (Rationalizing the Denominator)
To divide radical expressions with the same index, we use the quotient rule for radicals. If a and b represent nonnegative numbers, where
b
≠
0
, then we have
Example 10: Divide:
√
80
√
10
.
Solution: In this case, we can see that 10 and 80 have common factors. If we apply the quotient rule for radicals and write it as a single square root, we will be able to reduce the fractional radicand.
Answer:
2
√
2
Example 11: Divide:
√
16
x
5
y
4
√
2
x
y
.
Solution:
Answer:
2
x
2
y
√
2
y
Example 12: Divide:
3
√
54
a
3
b
5
3
√
16
a
2
b
2
.
Solution:
Answer:
3
b
⋅
3
√
a
2