Simplify (3x ⁄ 2y)(x ⁄ 3y)
Answer: D
Multiply numerators and denominators separately, then cancel.
\[ \frac{3x}{2y} \cdot \frac{x}{3y} = \frac{3x^{2}}{6y^{2}} = \frac{x^{2}}{2y^{2}} \]
The 3 in the numerator cancels one factor of the 6 - leaving it in gives \( \tfrac{x^{2}}{6y^{2}} \).
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On simplifying expression 13y + 2z + 3y, answer will be
Answer: B
Only the \( y \) terms are alike.
\[ 13y + 3y = 16y \]
\[ \therefore\ 16y + 2z \]
The \( z \) term is unlike and stays as it is; it cannot be merged with the \( y \) terms.
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Product of 4 and -3b is
Answer: B
\[ 4 \times (-3b) = -12b \]
A positive times a negative is negative, and the numbers multiply as 4 and 3 to give 12.
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By factorizing 54x - 45y, answer must be
Answer: A
Both terms share a factor of 9.
\[ 54x - 45y = 9 \times 6x - 9 \times 5y = 9(6x - 5y) \]
9 is the highest common factor of 54 and 45; taking out 3 alone would leave \( 3(18x - 15y) \), which is not fully factorised.
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By factorizing following expression 5ab + 10ac + 15ad, answer must be
Answer: B
Every term shares the factor \( 5a \).
\[ 5ab + 10ac + 15ad = 5a(b + 2c + 3d) \]
Checking: \( 5a \times b = 5ab \), \( 5a \times 2c = 10ac \), \( 5a \times 3d = 15ad \). Taking out 5 alone would leave the \( a \) behind in each term.
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If algebraic expression is [4(a + 2b - 3c) - 3(2b - 3c)] + 2a then answer will be
Answer: B
Expand each bracket, then collect.
\[ 4(a + 2b - 3c) = 4a + 8b - 12c \]
\[ -3(2b - 3c) = -6b + 9c \]
\[ (4a + 8b - 12c) + (-6b + 9c) + 2a = 6a + 2b - 3c \]
The \( -3 \) turns \( -3c \) into \( +9c \), which is what leaves only \( -3c \) at the end rather than \( -21c \).
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Simplify (1 ⁄ (x² - 1)) ÷ (1 ⁄ (x + 1))
Answer: A
Dividing by a fraction means multiplying by its reciprocal.
\[ \frac{1}{x^{2}-1} \div \frac{1}{x+1} = \frac{1}{x^{2}-1} \times (x+1) \]
Factorising the denominator:
\[ = \frac{x+1}{(x-1)(x+1)} = \frac{1}{x-1} \]
The \( (x+1) \) cancels, which is why the difference of squares has to be split before the simplification becomes visible.
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Expand and simplify (x + y)³
Answer: B
\[ (x+y)^{3} = x^{3} + 3x^{2}y + 3xy^{2} + y^{3} \]
The two middle terms share a factor of \( 3xy \):
\[ 3x^{2}y + 3xy^{2} = 3xy(x + y) \]
\[ \therefore\ (x+y)^{3} = x^{3} + 3xy(x+y) + y^{3} \]
The sign inside the bracket must match the original: \( (x-y)^{3} = x^{3} - 3xy(x-y) - y^{3} \).
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By factorizing expression 2bx + 4by - 3ax -6ay, answer must be
Answer: B
Group the terms in pairs and factor each.
\[ 2bx + 4by - 3ax - 6ay = 2b(x + 2y) - 3a(x + 2y) \]
The bracket \( (x + 2y) \) is now common:
\[ = (2b - 3a)(x + 2y) \]
Taking out \( -3a \) rather than \( 3a \) from the second pair is what makes both brackets identical.
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Algebraic expression of sum of 10a and 15c can be written as
Answer: A
"Sum" means addition, and the two terms are unlike - one in \( a \), the other in \( c \) - so they cannot be combined.
\[ 10a + 15c \]
Only like terms may be added; \( 25ac \) would be the result of a quite different operation.
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