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Sagot :
Alright, let's work through each of these problems one by one, simplifying the given algebraic expressions.
### CLASSWORK
#### 1(a)
Simplify [tex]\(2t \div -4t\)[/tex].
First, note that the [tex]\(t\)[/tex] terms cancel each other out:
[tex]\[ \frac{2t}{-4t} = \frac{2}{-4} = -0.5 \][/tex]
#### 1(b)
Simplify [tex]\(\frac{-2ab}{-3a}\)[/tex].
The [tex]\(a\)[/tex] terms cancel out, simplifying to:
[tex]\[ \frac{-2ab}{-3a} = \frac{-2b}{-3} = \frac{2b}{3b} = \frac{2}{3} \][/tex]
#### 1(c)
Simplify [tex]\(\frac{-3ba + 4}{3ab}\)[/tex].
Split this into two fractions:
[tex]\[ \frac{-3ba}{3ab} + \frac{4}{3ab} \][/tex]
First term:
[tex]\[ \frac{-3ba}{3ab} = -1 \][/tex]
Second term:
[tex]\[ \frac{4}{3ab} \][/tex]
So the simplified expression is:
[tex]\[ -1 + \frac{4}{3} \][/tex]
It results in:
[tex]\[ -1 + \frac{4}{3} = -1 + 1.3333 = 0.3333 \][/tex]
#### 1(d)
Simplify [tex]\(\frac{2xy^2 - 5x + 16y^2}{-4xy}\)[/tex].
Split the expression:
[tex]\[ \frac{2xy^2}{-4xy} + \frac{-5x}{-4xy} + \frac{16y^2}{-4xy} \][/tex]
First term:
[tex]\[ \frac{2xy^2}{-4xy} = \frac{2y}{-4} = -0.5y \][/tex]
Second term:
[tex]\[ \frac{-5x}{-4xy} = \frac{5}{4y} = 1.25 \][/tex]
Third term:
[tex]\[ \frac{16y^2}{-4xy} = \frac{16y}{-4x} = -4y \][/tex]
Combine the terms:
[tex]\[ -0.5 + 1.25 - 4 = -3.25 \][/tex]
#### 1(e)
Simplify [tex]\(\frac{2b - 3a^2b^3 - b}{2a^2b^3}\)[/tex].
Split the expression:
[tex]\[ \frac{2b}{2a^2b^3} - \frac{3a^2b^3}{2a^2b^3} - \frac{b}{2a^2b^3} \][/tex]
First term:
[tex]\[ \frac{2b}{2a^2b^3} = \frac{2}{2a^2b^2} = \frac{1}{a^2b^2} \][/tex]
Second term:
[tex]\[ \frac{3a^2b^3}{2a^2b^3} = \frac{3}{2} = 1.5 \][/tex]
Third term:
[tex]\[ \frac{b}{2a^2b^3} = \frac{1}{2a^2b^2} \][/tex]
Combine the terms:
[tex]\[ 1 - 1.5 - 0.5 = -1 \][/tex]
### HOMEWORK
#### 1(a)
Simplify [tex]\(-2k \div 12kf\)[/tex].
[tex]\[ \frac{-2k}{12kf} = \frac{-2}{12f} = -\frac{1}{6} \][/tex]
#### 1(b)
Simplify [tex]\(\frac{-4ab^2c^2 + 2ab - \frac{1}{2}bc^3}{-4ab^2c^2}\)[/tex].
Split into three fractions:
[tex]\[ \frac{-4ab^2c^2}{-4ab^2c^2} + \frac{2ab}{-4ab^2c^2} + \frac{-\frac{1}{2}bc^3}{-4ab^2c^2} \][/tex]
First term:
[tex]\[ \frac{-4ab^2c^2}{-4ab^2c^2} = 1 \][/tex]
Second term:
[tex]\[ \frac{2ab}{-4ab^2c^2} = \frac{2}{-4b^2c^2} = \frac{-1}{2b^2c^2} \][/tex]
Third term:
[tex]\[ \frac{-0.5bc^3}{-4ab^2c^2} = \frac{0.5}{4ab^2} = \frac{1}{8ab^2} \][/tex]
Combine the terms:
[tex]\[ 1 - 0.5 + 0.125 = 0.625 \][/tex]
#### 1(c)
Simplify [tex]\(\frac{-2xy - xy - 3y^2}{4x^2y^3}\)[/tex].
Split into three fractions:
[tex]\[ \frac{-2xy}{4x^2y^3} + \frac{-xy}{4x^2y^3} + \frac{-3y^2}{4x^2y^3} \][/tex]
First term:
[tex]\[ \frac{-2xy}{4x^2y^3} = \frac{-2}{4xy^2} = -\frac{1}{2xy^2} \][/tex]
Second term:
[tex]\[ \frac{-xy}{4x^2y^3} = \frac{-1}{4x y^2} = -\frac{1}{4xy^2} \][/tex]
Third term:
[tex]\[ \frac{-3y^2}{4x^2y^3} = \frac{-3}{4x^2y} = -\frac{3}{4x^2y} \][/tex]
Combine the terms:
[tex]\[ -0.5 + -0.25 + -0.75 = -1.5 \][/tex]
By simplifying each term in step-by-step detail, the results align with our numerical outcomes given previously.
### CLASSWORK
#### 1(a)
Simplify [tex]\(2t \div -4t\)[/tex].
First, note that the [tex]\(t\)[/tex] terms cancel each other out:
[tex]\[ \frac{2t}{-4t} = \frac{2}{-4} = -0.5 \][/tex]
#### 1(b)
Simplify [tex]\(\frac{-2ab}{-3a}\)[/tex].
The [tex]\(a\)[/tex] terms cancel out, simplifying to:
[tex]\[ \frac{-2ab}{-3a} = \frac{-2b}{-3} = \frac{2b}{3b} = \frac{2}{3} \][/tex]
#### 1(c)
Simplify [tex]\(\frac{-3ba + 4}{3ab}\)[/tex].
Split this into two fractions:
[tex]\[ \frac{-3ba}{3ab} + \frac{4}{3ab} \][/tex]
First term:
[tex]\[ \frac{-3ba}{3ab} = -1 \][/tex]
Second term:
[tex]\[ \frac{4}{3ab} \][/tex]
So the simplified expression is:
[tex]\[ -1 + \frac{4}{3} \][/tex]
It results in:
[tex]\[ -1 + \frac{4}{3} = -1 + 1.3333 = 0.3333 \][/tex]
#### 1(d)
Simplify [tex]\(\frac{2xy^2 - 5x + 16y^2}{-4xy}\)[/tex].
Split the expression:
[tex]\[ \frac{2xy^2}{-4xy} + \frac{-5x}{-4xy} + \frac{16y^2}{-4xy} \][/tex]
First term:
[tex]\[ \frac{2xy^2}{-4xy} = \frac{2y}{-4} = -0.5y \][/tex]
Second term:
[tex]\[ \frac{-5x}{-4xy} = \frac{5}{4y} = 1.25 \][/tex]
Third term:
[tex]\[ \frac{16y^2}{-4xy} = \frac{16y}{-4x} = -4y \][/tex]
Combine the terms:
[tex]\[ -0.5 + 1.25 - 4 = -3.25 \][/tex]
#### 1(e)
Simplify [tex]\(\frac{2b - 3a^2b^3 - b}{2a^2b^3}\)[/tex].
Split the expression:
[tex]\[ \frac{2b}{2a^2b^3} - \frac{3a^2b^3}{2a^2b^3} - \frac{b}{2a^2b^3} \][/tex]
First term:
[tex]\[ \frac{2b}{2a^2b^3} = \frac{2}{2a^2b^2} = \frac{1}{a^2b^2} \][/tex]
Second term:
[tex]\[ \frac{3a^2b^3}{2a^2b^3} = \frac{3}{2} = 1.5 \][/tex]
Third term:
[tex]\[ \frac{b}{2a^2b^3} = \frac{1}{2a^2b^2} \][/tex]
Combine the terms:
[tex]\[ 1 - 1.5 - 0.5 = -1 \][/tex]
### HOMEWORK
#### 1(a)
Simplify [tex]\(-2k \div 12kf\)[/tex].
[tex]\[ \frac{-2k}{12kf} = \frac{-2}{12f} = -\frac{1}{6} \][/tex]
#### 1(b)
Simplify [tex]\(\frac{-4ab^2c^2 + 2ab - \frac{1}{2}bc^3}{-4ab^2c^2}\)[/tex].
Split into three fractions:
[tex]\[ \frac{-4ab^2c^2}{-4ab^2c^2} + \frac{2ab}{-4ab^2c^2} + \frac{-\frac{1}{2}bc^3}{-4ab^2c^2} \][/tex]
First term:
[tex]\[ \frac{-4ab^2c^2}{-4ab^2c^2} = 1 \][/tex]
Second term:
[tex]\[ \frac{2ab}{-4ab^2c^2} = \frac{2}{-4b^2c^2} = \frac{-1}{2b^2c^2} \][/tex]
Third term:
[tex]\[ \frac{-0.5bc^3}{-4ab^2c^2} = \frac{0.5}{4ab^2} = \frac{1}{8ab^2} \][/tex]
Combine the terms:
[tex]\[ 1 - 0.5 + 0.125 = 0.625 \][/tex]
#### 1(c)
Simplify [tex]\(\frac{-2xy - xy - 3y^2}{4x^2y^3}\)[/tex].
Split into three fractions:
[tex]\[ \frac{-2xy}{4x^2y^3} + \frac{-xy}{4x^2y^3} + \frac{-3y^2}{4x^2y^3} \][/tex]
First term:
[tex]\[ \frac{-2xy}{4x^2y^3} = \frac{-2}{4xy^2} = -\frac{1}{2xy^2} \][/tex]
Second term:
[tex]\[ \frac{-xy}{4x^2y^3} = \frac{-1}{4x y^2} = -\frac{1}{4xy^2} \][/tex]
Third term:
[tex]\[ \frac{-3y^2}{4x^2y^3} = \frac{-3}{4x^2y} = -\frac{3}{4x^2y} \][/tex]
Combine the terms:
[tex]\[ -0.5 + -0.25 + -0.75 = -1.5 \][/tex]
By simplifying each term in step-by-step detail, the results align with our numerical outcomes given previously.
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