Fractional and negative indices

(a) $5$

1. A power of $\frac{1}{2}$ is a square root: $\sqrt{25} = 5$.

• B1 The correct answer, $5$.

(b) $\frac{1}{8}$

1. A negative power is a reciprocal: $8^{-1} = \frac{1}{8}$.

• B1 The correct answer, $\frac{1}{8}$.

(c) $9$

1. Take the cube root first, then square.
   $$27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$$

• M1 Finding $\sqrt[3]{27} = 3$.
• A1 The correct answer, $9$.

(d) $\frac{3}{2}$

1. The negative index flips the fraction: $\left(\frac{9}{4}\right)^{\frac{1}{2}}$.
2. The square root of each part: $\frac{3}{2}$.

• M1 Flipping to $\frac{9}{4}$ or square-rooting to $\frac{2}{3}$.
• A1 The correct answer, $\frac{3}{2}$.

(a) $8$

1. Flip for the negative index.
   $$\left(\frac{1}{16}\right)^{-\frac{3}{4}} = 16^{\frac{3}{4}}$$
2. Fourth root, then cube.
   $$\left(\sqrt[4]{16}\right)^3 = 2^3 = 8$$

• M1 Flipping to $16^{\frac{3}{4}}$, or finding $\sqrt[4]{16} = 2$.
• A1 The correct answer, $8$.

(b) $0.04$

1. $5^{-2} = \frac{1}{25} = 0.04$.

• B1 The correct answer, $0.04$.

(a) $x^{-\frac{1}{2}}$

1. $\sqrt{x} = x^{\frac{1}{2}}$, and one over it is $x^{-\frac{1}{2}}$.

• B1 $x^{-\frac{1}{2}}$.

(b) $8x^6$

1. Apply the power to each factor.
   $$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8, \qquad \left(x^8\right)^{\frac{3}{4}} = x^6$$

• M1 One part correct: $8$ or $x^6$.
• A1 The correct answer, $8x^6$.

(a) $\frac{2}{3}$

1. $$16^{3/4}$$
2. $$8^{-2/3}$$
3. $$8\times 1/4/3$$
4. 16^3/4 = (fourth root of 16)$^{3}$ = $2^{3}$ = 8.
5. 8^($-2$/3) = 1/(cube root of 8)$^{2}$ = 1/4, and 27^1/3 = 3.
6. So the value is 8 $\times$ 1/4 $\div$ 3 = 2/3.

• M1 Establishing $16^{3/4}$ or an equivalent valid method.
• M1 Establishing $8^{-2/3}$ or an equivalent valid method.
• M1 Establishing $8\times 1/4/3$ or an equivalent valid method.
• A1 Correct answer: $\frac{2}{3}$

(a) $\frac{9}{4}$

1. Take the cube root before squaring.
   $$(27^{1/3})^{2}$$
2. A negative half power is the reciprocal square root.
   $$9/4$$
3. Therefore $\frac{9}{4}$.

• M1 Take the cube root before squaring.
• M1 A negative half power is the reciprocal square root.
• A1 Correct answer: $\frac{9}{4}$

(b) $-3$

1. Write the reciprocal as a negative integer power.
   $$3^{-3}$$
2. Therefore $-3$.

• M1 Write the reciprocal as a negative integer power.
• A1 Correct answer: $-3$

(a) $4$

1. Evaluate the positive fractional power.
   $$16^{3/2}$$
2. Multiply by the reciprocal of the squared cube root.
   $$64/16$$
3. Therefore $4$.

• M1 Evaluate the positive fractional power.
• M1 Multiply by the reciprocal of the squared cube root.
• A1 Correct answer: $4$

(b) $1024$

1. Dividing by the reciprocal is multiplication.
   $$64\times 16$$
2. Therefore $1024$.

• M1 Dividing by the reciprocal is multiplication.
• A1 Correct answer: $1024$

(a) $2$

1. Write 8 as a power of 2.
   $$(2^{3})^{2/3}$$
2. Therefore $2$.

• M1 Write 8 as a power of 2.
• A1 Correct answer: $2$

(b) $-\frac{5}{2}$

1. Write the square root and the reciprocal as powers.
   $$(2^{5})^{-1/2}$$
2. Therefore $-\frac{5}{2}$.

• M1 Write the square root and the reciprocal as powers.
• A1 Correct answer: $-\frac{5}{2}$

(c) $-\frac{9}{4}$

1. Write every term as a power of 2 and equate the powers.
   $$2+2x=-5/2$$
2. Therefore $-\frac{9}{4}$.

• M1 Write every term as a power of 2 and equate the powers.
• A1 Correct answer: $-\frac{9}{4}$

(a) $x = \frac{7}{3}$

1. Write every term as a power of 2.
   $$4^x = 2^{2x}, \qquad 32^2 = \left(2^5\right)^2 = 2^{10}$$
2. Add the indices on the left.
   $$2^{2x + x + 3} = 2^{10} \Rightarrow 3x + 3 = 10$$
3. $$x = \frac{7}{3}$$

• P1 Writing $4^x$ as $2^{2x}$.
• P1 Writing $32^2$ as $2^{10}$.
• P1 Forming $3x + 3 = 10$.
• A1 $x = \frac{7}{3}$.

(b) $n = -\frac{5}{3}$

1. Powers of 3.
   $$9^{\frac{1}{2}} = 3, \qquad 27^n = 3^{3n}$$
2. $$3^{1 + 3n} = 3^{-4} \Rightarrow 1 + 3n = -4 \Rightarrow n = -\frac{5}{3}$$

• P1 Writing $9^{\frac{1}{2}} = 3^1$ and $27^n = 3^{3n}$.
• A1 $n = -\frac{5}{3}$.