Year 8 Algebraic techniques and index laws: free practice test with answers

Sample algebraic techniques and index laws questions with answers

Try each one before you open the answer.

1. The language of algebra

   For the expression $5x - 3y + 7 - x$:

   (a) How many terms are there?

   (b) What is the coefficient of $y$?

   (c) What is the constant term?

   Show the answer

   Key idea: Terms are separated by $+$ and $-$ signs, and each term carries the sign in front of it.

   1. (a) The terms are $5x$, $-3y$, $7$ and $-x$, so there are $4$ terms
   2. (b) The $y$ term is $-3y$, so the coefficient is $-3$
   3. (c) The term with no pronumeral is $7$

   Answer: (a) $4$ (b) $-3$ (c) $7$

2. Substitution and equivalence

   Evaluate $3a^2 - 2b$ when $a = -2$ and $b = 5$.

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   Key idea: Substitute using brackets around negative numbers, then follow the order of operations.

   1. $3 \times (-2)^2 - 2 \times 5$
   2. $= 3 \times 4 - 10$
   3. $= 12 - 10 = 2$

   Answer: $2$

3. Adding and subtracting terms

   Simplify $7ab - 3ba + 2a - 5a$.

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   Key idea: Like terms have exactly the same pronumerals; $ab$ and $ba$ are like terms.

   1. $7ab - 3ba = 7ab - 3ab = 4ab$
   2. $2a - 5a = -3a$
   3. So the expression simplifies to $4ab - 3a$

   Answer: $4ab - 3a$

4. Adding and subtracting algebraic fractions

   Simplify $\dfrac{x}{4} + \dfrac{x}{6}$.

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   Key idea: Write both fractions with the lowest common denominator, then add the numerators.

   1. The LCD of $4$ and $6$ is $12$
   2. $\dfrac{3x}{12} + \dfrac{2x}{12} = \dfrac{5x}{12}$

   Answer: $\dfrac{5x}{12}$

5. Multiplying and dividing algebraic fractions

   Simplify $\dfrac{2x}{5} \times \dfrac{15}{4x}$.

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   Key idea: Cancel common factors between any numerator and any denominator, then multiply.

   1. Cancel $x$, then $15$ with $5$ (leaving $3$ and $1$) and $2$ with $4$ (leaving $1$ and $2$)
   2. $\dfrac{1}{1} \times \dfrac{3}{2} = \dfrac{3}{2}$

   Answer: $\dfrac{3}{2}$

6. Expanding brackets

   Expand and simplify $3(2x + 5) - 4x + 7$.

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   Key idea: Use the distributive law to expand the bracket first, then collect like terms.

   1. Expand the bracket: $3(2x + 5) = 6x + 15$
   2. Rewrite the full expression: $6x + 15 - 4x + 7$
   3. Collect the $x$ terms: $6x - 4x = 2x$
   4. Collect the constant terms: $15 + 7 = 22$
   5. Write the simplified answer: $2x + 22$

   Answer: $2x + 22$

7. Applying algebra

   A school canteen sells pies for $p$ dollars each and drinks for $d$ dollars each. On Monday, Aisha buys 3 pies and 2 drinks. On Tuesday, she buys 1 pie and 4 drinks. Write a single simplified expression for her total spending over both days. Then find the total when $p = 4$ and $d = 2.50$.

   Show the answer

   Key idea: Write an expression for each day, add them, collect like terms, then substitute.

   1. Monday's cost: $3p + 2d$
   2. Tuesday's cost: $p + 4d$
   3. Total: $(3p + 2d) + (p + 4d) = 3p + p + 2d + 4d$
   4. Simplify: $4p + 6d$
   5. Substitute $p = 4$ and $d = 2.50$: $4(4) + 6(2.50) = 16 + 15 = 31$

   Answer: The simplified expression is $4p + 6d$; the total is $31.

8. Index laws for multiplication and division

   Simplify each expression. Give your answer in exponent form.

   (a) $3^4 \times 3^5$

   (b) $\dfrac{2^9}{2^3}$

   Show the answer

   Key idea: When multiplying powers with the same base, add the exponents. When dividing, subtract the exponents.

   1. (a) $3^4 \times 3^5 = 3^{4+5} = 3^9$
   2. (b) $\dfrac{2^9}{2^3} = 2^{9-3} = 2^6$

   Answer: (a) $3^9$ (b) $2^6$

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 8 algebra worksheet
• Year 8 exponents worksheet