Like terms and algebraic index rules

(a) $5m + 8n$

1. $$7m-2m+3n+5n$$
2. Collect the m terms: 7m $-$ 2m = 5m.
3. Collect the n terms: 3n + 5n = 8n.

• M1 Establishing $7m-2m+3n+5n$ or an equivalent valid method.
• A1 Correct answer: $5m + 8n$

(a) $3x + 4y$

1. Collect the $x$ terms: $5x - 2x = 3x$. Collect the $y$ terms: $3y + y = 4y$.

• B1 One correct term, $3x$ or $4y$.
• B1 The correct answer, $3x + 4y$.

(b) $p^3$

1. Three $p$s multiplied together: $p^3$.

• B1 The correct answer, $p^3$.

(c) $12ab$

1. Multiply the numbers and write the letters together: $4 \times 3 = 12$, so $12ab$.

• B1 The correct answer, $12ab$.

(a) $x^8$

1. Add the indices: $5 + 3 = 8$.

• B1 The correct answer, $x^8$.

(b) $4a^6$

1. Square the 2 and the $a^3$: $2^2 = 4$ and $(a^3)^2 = a^6$.

• B1 One part correct: $4$ or $a^6$.
• B1 The correct answer, $4a^6$.

(c) $3x^4y$

1. Divide the numbers and subtract the indices of each letter.
2. $$\frac{12}{4} = 3, \quad x^{7 - 3} = x^4, \quad y^{2 - 1} = y$$

• M1 Two of the three parts correct: $3$, $x^4$, $y$.
• A1 The correct answer, $3x^4y$.

(a) $10x+10$

1. Add two copies of each side and collect like terms.
   $$2(3x+2)+2(2x+3)$$
2. Therefore $10x+10$.

• M1 Add two copies of each side and collect like terms.
• A1 Correct answer: $10x+10$

(b) $30$ cm

1. Substitute into the simplified perimeter.
   $$10\times 2+10$$
2. Therefore $30$ cm.

• M1 Substitute into the simplified perimeter.
• A1 Correct answer: $30$ cm

(a) $2x^{3}y^{2}$

1. Divide the coefficients and subtract powers of each matching base.
   $$\frac{6}{3}x^{5-2}y^{3-1}$$
2. Therefore $2x^{3}y^{2}$.

• M1 Divide the coefficients and subtract powers of each matching base.
• A1 Correct answer: $2x^{3}y^{2}$

(b) $-144$

1. Keep brackets around the negative base raised to an odd power.
   $$2\times (-2)^{3}\times 3^{2}$$
2. Therefore $-144$.

• M1 Keep brackets around the negative base raised to an odd power.
• A1 Correct answer: $-144$

(a) $a+9b$

1. Double every term of E and subtract every term of F.
   $$2(4a+4b-2a)-(3a-b)$$
2. Therefore $a+9b$.

• M1 Double every term of E and subtract every term of F.
• A1 Correct answer: $a+9b$

(b) $-7$

1. Substitute into the collected expression.
   $$1\times 2+9\times (-1)$$
2. Therefore $-7$.

• M1 Substitute into the collected expression.
• A1 Correct answer: $-7$

(a) $2x^{7}y^{5}$

1. $$8x^{9}y^{6}/(4x^{2}y)$$
2. Raise each factor to the third power: ($2x^{3}y^{2}$)$^{3}$ = $8x^{9}y^{6}.$
3. When dividing equal bases, subtract powers: $8x^{9}y^{6}$/($4x^{2}y$) = $2x^{7}y^{5}.$

• M1 Establishing $8x^{9}y^{6}/(4x^{2}y)$ or an equivalent valid method.
• A1 Correct answer: $2x^{7}y^{5}$

(a) $\frac{27x^6y^3}{9xy^2} = 3x^5y$

1. Cube each part of the bracket.
   $$(3x^2y)^3 = 27x^6y^3$$
2. Divide.
   $$\frac{27x^6y^3}{9xy^2} = 3x^{6 - 1}y^{3 - 2} = 3x^5y$$

• M1 Expanding the numerator to $27x^6y^3$.
• A1 Dividing correctly to reach $3x^5y$ with each index shown.

(b) $y = 3$

1. Write $4^y$ as a power of 2.
   $$4^y = (2^2)^y = 2^{2y}, \quad 2^x \times 2^{2y} = 2^{x + 2y}$$
2. So $x + 2y = 10$.
3. Subtract $x + y = 7$.
   $$(x + 2y) - (x + y) = 10 - 7 \Rightarrow y = 3$$

• P1 Writing $4^y$ as $2^{2y}$.
• P1 Forming the equation $x + 2y = 10$.
• P1 Solving with $x + y = 7$ (subtracting or substituting).
• A1 The correct answer, $y = 3$.