Download Section 11-1: Irrational Numbers and Real Numbers Section 11

Section 11-1: Irrational Numbers and Real Numbers
Learning Outcome 1
Write in rational exponent and radical notation.
Square root of 9.
Cube root of 64.
1
3
1
2
1
64 = 3 64 = 4
9 = 9 =3
Fourth root of 16.
16 4 = 4 16 = 2
Learning Outcome 2
1
1
7 is between 4 and 9, so 7 2 is between 2 and 3. 24 is between 16 and 25, so 24 2 is between 4
and 5.
Learning Outcome 3
1
2
1
2
1
2
11 = 3.31662479; 12 = 3.464101615; 13 = 3.605551275
Learning Outcome 4
Write in fractional exponent form:
7
5
4
=5
4
7
The index (root) is the denominator of the fractional exponent.
Write in radical form:
7
3
3
5 = 157
The denominator is the index (root) of the radical.
Section 11-2: Simplifying Irrational Expressions
Learning Outcome 1
x 14 ;
Find the square root:
x 14 = x 7 ;
x 42 = x 21
x 42
To find the square root of variable factors, divide the exponent by 2.
Learning Outcome 2
Simplify:
1
4
3
4
1
4
3
4
x •x
x •x
=x
1 3
+
4 4
=x
4
4
1
=x =x
In multiplication add exponents when bases are alike.
3
8
1
2
3
8
1
2
x •x
x •x
x
x
7
16
7
16
÷x
÷x
=x
3 1
+
8 2
=x
3 4
+
8 8
=x
7
8
In multiplication add exponents when bases are alike.
Get common denominators when necessary.
1
4
1
4
=x
7 1
−
16 4
=x
7 4
−
16 16
=x
3
16
In division subtract exponents when bases are alike.
Get common denominators when necessary.
Learning Outcome 3
Simplify the radical expressions.
63
Factor radicand so that one factor is the largest perfect square.
9 7
3 7
Take the square root of the perfect square factor.
x 17
Factor the radicand so that one factor is a perfect square (with an
even-numbered exponent).
x 16 x
Take the square root of the perfect square factor by taking one-half
the exponent.
x8 x
90x 7
6
Factor each factor of radicand for perfect squares.
9 x 10 x
3 x 3 10 x
Take the square root of each perfect square factor.
4 24
Factor radicand for perfect squares (do not factor coefficient).
4 4•6
Take the square root of the perfect square factor.
4•2 6
8 6
Multiply factors in the coefficient (outside the radical).
Section 11-3: Basic Operations with Square-Root Radicals
Learning Outcome 1
Add or subtract the radical expressions.
4 3x + 6 3x
Because the radicals are like radicals, add the coefficients.
10 3 x
13 63 − 8 28
Simplify radicals before attempting to subtract.
13 9 • 7 − 8 4 • 7
Factor perfect squares under the radical and take square roots.
13 • 3 7 − 8 • 2 7
Multiply coefficients for each term.
39 7 − 16 7
23 7
Subtract coefficients.
Learning Outcome 2
Multiply and simplify.
6 10 • 4 15
24 150
Multiply the coefficients of each radical and write the product
outside the radical. Multiply the radicands and write the product
inside the radical.
Factor the radicand for the largest perfect square factor.
24 25 • 6
Write the perfect square of 25 outside the radicand.
24 • 5 6
120 6
Multiply the coefficients.
Learning Outcome 3
Divide.
15 48
12 28
5 16 • 3
Reduce coefficients and simplify radicals.
Simplify radicals.
4 4•7
5•4 3
Multiply or reduce coefficients.
4•2 7
5 3
This expression is not completely simplified until the radical has
2 7
been removed from the denominator (see Learning Outcome 4,
which follows.)
Learning Outcome 4
Rationalize the denominator.
10
Rewrite the radical as two separate fractions and multiply by a
3
factor that will make the radical in the denominator a perfect square.
3
10
10
10
3
Take square root of denominator. Multiplying by
is equivalent
=
=
•
3
3
3
3
3
to multiplying by 1.
30
9
=
30
3
Can 30 and 3 be reduced? No! the 30 is under the radical.
Section 11-4: Complex and Imaginary Numbers
Learning Outcome 1
Simplify: − 63
− 63 = − 1(63)
Factor –63 using –1 as one factor and a perfect square as another.
− 1(9)(7)
Write each factor as a square root.
− 1 = i ; 9 = 3 ; Simplify expression.
−1 9 7
3i 7
Learning Outcome 2
Use the fact that
− 1 = i to simplify: i 29
i 29 = i 28 i 1
Factor so that one exponent is the greatest possible multiple of 4.
i raised to an exponent that is a multiple of 4 is equal to 1.
i 28 i 1 = i 1 = i
i 28 = 1
Learning Outcome 3
Write the following in complex number form: a + bi.
56
56 + 0i
Because 56 is a real number, the imaginary part of the complex
number is zero.
− 37
0 + i 37
Learning Outcome 4
Simplify: (7 + 3i) − (12 − i)
7 + 3i − 12 + i
7 − 12 + (3i + i)
–5 + 4i
−5 + 4i
Because the real part of the number is zero, we write the imaginary
part of the number with i and the real part of the number as zero.
Distribute to remove the second set of grouping symbols.
Regroup so that real parts are grouped together and imaginary
parts are grouped together.
Combine the real parts and combine the imaginary parts.
Section 11-5: Equations with Squares and Square Roots
Learning Outcome 1
Solve the equation for x: 5x2 − 7 = 40
5x2 − 7 = 40
Add –7 to both sides of the equation.
5x2 = 47
Divide both sides of equation by 5.
47
x2 =
Take square root of both sides.
5
47
5
235
x=±
5
x = ± 3.066
x=±
Simplify expression.
Exact solutions.
Approximate solutions.
Learning Outcome 2
Solve the equation containing a radical:
5 x − 8 = 10
5x − 8 = 100
5x = 108
x=
5 x − 8 = 10
Be sure radical expression is isolated on one side of equation.
Square both sides of the equation.
Isolate the variable (letter) term.
Divide by the coefficient of x.
108
5
Learning Outcome 3
A right triangle has legs that measure 32 inches and 48 inches. What is the measure of the
hypotenuse?
In the Pythagorean theorem, c is the hypotenuse, and a and b are
c2 = a2 + b2
legs. Substitute numbers in place of letters.
c2 = 322 + 482
Simplify according to the order of operations. Square each number
first.
Perform addition, then take the square root.
c2 = 1,024 + 2,304
c = 57.69 inches