Integer powers and roots

(a) $6$

1. The cube root is the number whose cube is 216.
2. 6 $\times$ 6 $\times$ 6 = 216, so the cube root is 6.

• B1 Correct answer: $6$

(a) $64$

1. $4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$.

• B1 The correct answer, $64$.

(b) $9$

1. $9 \times 9 = 81$, so $\sqrt{81} = 9$.

• B1 The correct answer, $9$.

(c) $5$

1. $5 \times 5 \times 5 = 125$.

• B1 The correct answer, $5$.

(d) $2^5$

1. There are five 2s multiplied together, so it is $2^5$.

• B1 The correct answer, $2^5$.

(a) $5^{10}$

1. Multiplying powers of the same base: add the indices. $7 + 3 = 10$.

• B1 The correct answer, $5^{10}$.

(b) $7^5$

1. Dividing powers of the same base: subtract the indices. $9 - 4 = 5$.

• B1 The correct answer, $7^5$.

(c) $3^8$

1. A power of a power: multiply the indices. $2 \times 4 = 8$.

• B1 The correct answer, $3^8$.

(d) $n = 5$

1. Simplify each side as a power of 2.
   $$2^{n + 3} = 2^{8}$$
2. So $n + 3 = 8$, giving $n = 5$.

• M1 Writing the right-hand side as $2^8$, or the left-hand side as $2^{n+3}$.
• A1 The correct answer, $n = 5$.

(a) $32$

1. $$2^{8+3-6}$$
2. For equal bases, add powers when multiplying and subtract them when dividing.
3. $2^{8}$ $\times$ $2^{3}$ $\div$ $2^{6}$ = 2^(8 + 3 $-$ 6) = $2^{5}$ = 32.

• M1 Establishing $2^{8+3-6}$ or an equivalent valid method.
• A1 Correct answer: $32$

(a) $10$

1. A square requires even prime exponents, so add one factor each of 2 and 5.
   $$2\times 5$$
2. Therefore $10$.

• M1 A square requires even prime exponents, so add one factor each of 2 and 5.
• A1 Correct answer: $10$

(b) $60$

1. The completed square has even exponents.
   $$2^{4}\times 3^{2}\times 5^{2}$$
2. Halve each exponent to take its positive square root.
   $$2^{2}\times 3\times 5$$
3. Therefore $60$.

• M1 The completed square has even exponents.
• M1 Halve each exponent to take its positive square root.
• A1 Correct answer: $60$

(a) $6$

1. $$2^{5}\times 3^{3}\times 5^{2}$$
2. $$2\times 3$$
3. In the prime factorisation of a square, every exponent is even.
4. The exponents of 2 and 3 are odd, so divide by one factor of 2 and one factor of 3.
5. The smallest divisor that does this is k = 2 $\times$ 3 = 6; the quotient is $2^{4}$ $\times$ $3^{2}$ $\times$ $5^{2}.$

• P1 Establishing $2^{5}\times 3^{3}\times 5^{2}$ or an equivalent valid method.
• P1 Establishing $2\times 3$ or an equivalent valid method.
• A1 Correct answer: $6$

(a) $50$

1. $$540=2^{2}\times 3^{3}\times 5$$
2. $$2\times 5^{2}$$
3. Prime factorise: 540 = $2^{2}$ $\times$ $3^{3}$ $\times$ 5. A cube has prime exponents that are multiples of 3.
4. The smallest additions are one factor of 2 and two factors of 5.
5. k = 2 $\times$ $5^{2}$ = 50; then 540k = $2^{3}$ $\times$ $3^{3}$ $\times$ $5^{3}$ = $30^{3}.$

• P1 Establishing $540=2^{2}\times 3^{3}\times 5$ or an equivalent valid method.
• P1 Establishing $2\times 5^{2}$ or an equivalent valid method.
• A1 Correct answer: $50$

(a) $101250$

1. $$18=2\times 3^{2}$$
2. $$30=2\times 3\times 5$$
3. For the exponent b of 3, b $-$ 2 must be even and b $-$ 1 a multiple of 3. The smallest possible b is 4. For the exponent c of 5, c must be even and c $-$ 1 a multiple of 3, so the smallest c is 4.
4. $$2\times 3^{4}\times 5^{4}$$
5. Write 18 = 2 $\times$ $3^{2}$ and 30 = 2 $\times$ 3 $\times$ 5. For a square, prime exponents are even; for a cube, they are multiples of 3.
6. For the exponent a of 2 in n, a $-$ 1 must be even and a $-$ 1 must be a multiple of 3. The smallest possible a is 1.
7. For the exponent b of 3, b $-$ 2 must be even and b $-$ 1 a multiple of 3. The smallest possible b is 4. For the exponent c of 5, c must be even and c $-$ 1 a multiple of 3, so the smallest c is 4.
8. Thus n = 2 $\times$ $3^{4}$ $\times$ $5^{4}$ = 101250. Check: n/18 = 5625 = $75^{2}$ and n/30 = 3375 = $15^{3}.$

• P1 Establishing $18=2\times 3^{2}$ or an equivalent valid method.
• P1 Establishing $30=2\times 3\times 5$ or an equivalent valid method.
• P1 For the exponent b of 3, b $-$ 2 must be even and b $-$ 1 a multiple of 3. The smallest possible b is 4. For the exponent c of 5, c must be even and c $-$ 1 a multiple of 3, so the smallest c is 4.
• P1 Establishing $2\times 3^{4}\times 5^{4}$ or an equivalent valid method.
• A1 Correct answer: $101250$