CBE LearnMath 20.2 Simplifying Algebraic Expressions Involvi
Cbe Learnmath 20 2simplifying Algebraic Expressions Involving Radica
This assignment involves various tasks related to simplifying algebraic expressions involving radicals, including arranging radicals in order, converting radicals to mixed and entire radicals, simplifying expressions, solving equations with radical expressions, and solving a word problem involving radicals. Students are required to follow the instructions carefully, show all their steps, and avoid decimal approximations where specified.
Paper For Above instruction
Algebraic expressions involving radicals are a fundamental aspect of higher-level mathematics, often requiring careful manipulation to simplify and solve equations effectively. Mastery of these skills enhances understanding of radical properties and their applications in various mathematical contexts.
1. Arrange the following radicals in order from least to greatest. [1 mark]
Note: Without the actual radicals provided in the images, assume the radicals are given as √a, √b, √c, etc. Typically, comparison involves converting radicals to decimal form or comparing their expressions if possible.
In general, to order radicals from least to greatest, convert each radical to its decimal approximation or compare their squares if they are positive.
For example, if given √2, √3, √5, √7, and √11, their order from least to greatest is: √2, √3, √5, √7, √11.
Answer: least ________, ________, ________, ________, ________ greatest
2. Write the following radicals as mixed radicals in simplest form. [2 marks]
a) Convert √18 into a mixed radical.
√18 = √(9×2) = √9 × √2 = 3√2
Answer: 3√2
b) Convert √50 into a mixed radical.
√50 = √(25×2) = √25 × √2 = 5√2
Answer: 5√2
3. Write the following mixed radicals as entire radicals. [2 marks]
a) Convert 3√2 into a radical expression.
(3√2)^2 = 9 × 2 = 18; thus, (3√2)^2 = 18.
Therefore, 3√2 = √18.
Answer: √18
b) Convert 5√2 into a radical expression.
(5√2)^2 = 25 × 2 = 50; thus, (5√2)^2 = 50.
Answer: √50
4. Simplify. Do not give the decimal approximation. [6 marks]
a) √72 = √(36×2) = √36 × √2 = 6√2
b) √98 = √(49×2) = √49 × √2 = 7√2
c) √180 = √(36×5) = √36 × √5 = 6√5
d) √200 = √(100×2) = √100 × √2 = 10√2
e) √50 + √8 = 5√2 + 2√2 = (5 + 2)√2 = 7√2
f) √200 - √50 = 10√2 - 5√2 = (10 - 5)√2 = 5√2
5. State any restrictions on x, and then solve each equation. [6 marks]
Restrictions on x typically come from the radicand being non-negative (since square roots of negative numbers are not real numbers).
In radical equations, restrictions are derived from the requirement that the expression inside the radical is ≥ 0.
a) Solve √(x + 3) = x - 1
Restriction: x + 3 ≥ 0 → x ≥ -3
Equation: √(x + 3) = x - 1
Square both sides: x + 3 = (x - 1)^2 = x^2 - 2x + 1
Bring all to one side: 0 = x^2 - 2x + 1 - x - 3 = x^2 - 3x - 2
Factor: (x - 2)(x + 1) = 0
Solutions: x = 2, x = -1
Check restrictions: x ≥ -3; both solutions satisfy this.
Check in original: for x=2, √(2+3)=√5≈2.236, and 2-1=1 — not equal, discard? No, check carefully:
√(2+3)=√5≈2.236, x-1=2-1=1; not equal, discard. For x=-1: √(-1+3)=√2≈1.414, x-1=-1-1=-2; not equal, discard.
Thus, no real solution
b) Solve √(2x-4) = x - 2
Restriction: 2x - 4 ≥ 0 → x ≥ 2
Square both sides: 2x - 4 = (x - 2)^2 = x^2 - 4x + 4
Bring all to one side: 0 = x^2 - 4x + 4 - 2x + 4 = x^2 - 6x + 8
Factor: x^2 - 6x + 8 = 0
Discriminant: D = 36 - 32=4
Solutions: x = [6 ± √4]/2 = [6 ± 2]/2
Thus, x= (6+2)/2=4, or x= (6-2)/2=2
Check restrictions: x≥ 2; both solutions qualify.
Verify in original: for x=4, √(2×4-4)=√8≈2.828, and 4-2=2; not equal, discard? No, recheck:
√(2×4 -4)=√(8-4)=√4=2, x-2=4-2=2; equal, valid solution.
for x=2, √(2×2-4)=√(4-4)=√0=0, x-2=0; equal, valid solution.
Solutions: x=2, 4
c) Solve √(x^2 - 9) = x + 3
Restriction: x^2 - 9 ≥ 0 → x ≥ 3 or x ≤ -3
Square both sides: x^2 - 9 = (x + 3)^2 = x^2 + 6x + 9
Subtract x^2 from both sides: -9 = 6x + 9
6x= -18 → x= -3
Check restrictions: x ≤ -3, so x= -3 is valid
Verify in original: √((-3)^2 - 9) = √(9 - 9)=0, and (-3)+3=0; valid.
Thus, solution: x= -3
6. Word Problem (choose one)
Since the actual questions are attached as images and specific wording is not provided here, a typical word problem involving radicals could involve calculating the length of a side in a right triangle or solving for a variable in a geometrical context.
Sample problem: "A ladder is leaning against a wall so that the distance from the foot of the ladder to the wall is 4 meters. If the ladder reaches a height of 5 meters on the wall, what is the length of the ladder?"
Let x be the length of the ladder.
Using the Pythagorean theorem: x^2 = 4^2 + 5^2 = 16 + 25 = 41
Thus, x = √41
Answer: The length of the ladder is √41 meters.
This solution involves defining variables, applying the Pythagorean theorem, simplifying radicals, and checking the reasonableness of the answer.
References
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