If 1 n a and 1 n b are real numbers, then 1 n a # 1 n b = 1 n ab . Property Combining Radical Expressions: Products If 1 n a and 1 n b are real numbers and b ≠ 0, then 1 n a 1 n b = 5 n a b . Property Combining Radical Expressions: Quotients TEKS (7)(G) Rewrite radical expressions that contain variables to equivalent forms. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. Additional TEKS (1)(C), (1)(G) TEKS FOCUS • Rationalize the denominator – To rationalize the denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical. • Simplest form of a radical expression – The simplest form of a radical expression with index n is one where there are no nth power factors in any radical and the denominators are all rationalized. • Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated • Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data. VOCABULARY You can simplify a radical expression when the exponent of one factor of the radicand is a multiple of the radical’s index. You can simplify the product of powers that have the same exponent. Similarly, you can simplify the product of radicals that have the same index. ESSENTIAL UNDERSTANDING Same Exponent Same Index 2 3 = 2 3 2 2 3 2 = (2 3) 2 4 3 5 3 = (4 5) 3 4 5 = 4 5 3 3 3 9-2 Multiplying and Dividing Radical Expressions 393 PearsonTEXAS.com
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9-2 Multiplying and Dividing Radical Expressions · Multiplying Radical Expressions Can you simplify the product of the radical expressions? Explain. A 13 6 #12 No. The indexes are
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If 1n a and 1n b are real numbers, then 1n a # 1n b = 1n ab.
Property Combining Radical Expressions: Products
If 1n a and 1n b are real numbers and b ≠ 0, then 1n a
1n b= 5n a
b .
Property Combining Radical Expressions: Quotients
TEKS (7)(G) Rewrite radical expressions that contain variables to equivalent forms.
TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Additional TEKS (1)(C), (1)(G)
TEKS FOCUS
•Rationalize the denominator – To rationalize the denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical.
•Simplest form of a radical expression – The simplest form of a radical expression with index n is one where there are no nth power factors in any radical and the denominators are all rationalized.
•Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated
•Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.
VOCABULARY
You can simplify a radical expression when the exponent of one factor of the radicand is a multiple of the radical’s index.
You can simplify the product of powers that have the same exponent. Similarly, you can simplify the product of radicals that have the same index.
28. Apply Mathematics (1)(A) The formula t = 52sa shows the time t that any
vehicle takes to travel a distance s at a constant acceleration a, starting from rest. What is the difference in time between a car accelerating at 16 m>s2 and one accelerating at 25 m>s2 for a distance of 200 m?
29. The base of a triangle is 118 cm and its height is 18 cm. Find its area.
30. Use Representations to Communicate Mathematical Ideas (1)(E) The formula
F = mv2
r gives the centripetal force F of an object of mass m moving along a circle of radius r, where v is the tangential velocity of the object. Solve the formula for v. Rationalize the denominator.
31. Apply Mathematics (1)(A) The circular velocity v in miles per hour of a satellite
orbiting Earth is given by the formula v = 51.24 * 1012
r , where r is the distance in miles from the satellite to the center of Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 mi than the velocity of a satellite orbiting at an altitude of 200 mi? (The radius of Earth is 3950 mi.)
32. a. Simplify 22 + 23175
by multiplying the numerator and denominator by 175.
b. Simplify the expression in (a) by multiplying by 13 instead of 175.
c. Explain how you would simplify 22 + 23198
.
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396 Lesson 9-2 Multiplying and Dividing Radical Expressions
TEXAS Test Practice
49. What is the simplified form of the expression 3118xy 2
if x and y are positive?
A. 12x2xy B.
12y2xy C.
254xy 2
2xy D. 227xy 2
2xy
50. What are the solutions, in simplest form, of the quadratic equation 3x 2 + 6x - 5 = 0?
F. -6 { 1966 G. -6 { i124
6 H. -3 { 2163 J. -3 { i16
3
51. A triangle has the dimensions shown. What is the height of a triangle with equal area but a base of 36?
A. h3 B. 2h
3 C. 2h D. 3h
52. Find the axis of symmetry of the graph of the function y = -2x 2 - 5x + 4. Show your work.
h
12
Simplify each expression. Rationalize all denominators.