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Negative & fractional indices

1. ind-01 Present from here

   Work out $5^{-2}$

   A Option A: $-25$

   B Option B: $\frac{1}{25}$

   C Option C: $-10$

   D Option D: $\frac{1}{10}$
   Answer and misconceptions

   A Option A: $-25$ ✗

   Thinks a negative index makes the answer negativeA negative index means "one over": $5^{-2} = \frac{1}{5^2}$, which is positive.

   B Option B: $\frac{1}{25}$ ✓ Correct

   C Option C: $-10$ ✗

   Multiplies the base by the indexWorks out $5 \times (-2)$; an index tells you how many times to multiply the base by itself.

   D Option D: $\frac{1}{10}$ ✗

   Multiplies the base by the index, then takes the reciprocalWorks out $\frac{1}{5 \times 2}$; the reciprocal is right but $5^2 = 25$, not $10$.
2. ind-02 Present from here

   Work out $8^{\frac{1}{3}}$

   A Option A: $\frac{8}{3}$

   B Option B: $512$

   C Option C: $2\sqrt{2}$

   D Option D: $2$
   Answer and misconceptions

   A Option A: $\frac{8}{3}$ ✗

   Multiplies the base by the indexWorks out $8 \times \frac{1}{3}$; an index of $\frac{1}{3}$ means the cube root.

   B Option B: $512$ ✗

   Cubes instead of cube-rootingWorks out $8^3$; the $3$ is in the denominator, so it means the cube root: $\sqrt[3]{8} = 2$.

   C Option C: $2\sqrt{2}$ ✗

   Thinks any fractional index means square rootWorks out $\sqrt{8}$; the denominator $3$ means the cube root, not the square root.

   D Option D: $2$ ✓ Correct
3. ind-03 Present from here

   Work out $64^{\frac{2}{3}}$

   A Option A: $\frac{128}{3}$

   B Option B: $4$

   C Option C: $16$

   D Option D: $512$
   Answer and misconceptions

   A Option A: $\frac{128}{3}$ ✗

   Multiplies the base by the indexWorks out $64 \times \frac{2}{3}$; $64^{\frac{2}{3}}$ means the cube root of $64$, squared.

   B Option B: $4$ ✗

   Ignores the numerator of the indexFinds $\sqrt[3]{64} = 4$ but stops; the numerator $2$ means the result must then be squared: $4^2 = 16$.

   C Option C: $16$ ✓ Correct

   D Option D: $512$ ✗

   Swaps the roles of the numerator and denominatorTakes the square root and then cubes. The denominator gives the root (cube root) and the numerator gives the power (square).
4. ind-04 Present from here

   Work out $9^{-\frac{1}{2}}$

   A Option A: $\frac{1}{3}$

   B Option B: $-3$

   C Option C: $3$

   D Option D: $-\frac{9}{2}$
   Answer and misconceptions

   A Option A: $\frac{1}{3}$ ✓ Correct

   B Option B: $-3$ ✗

   Thinks a negative index makes the answer negativeFinds $\sqrt{9} = 3$ and makes it negative; the negative index means the reciprocal: $\frac{1}{3}$.

   C Option C: $3$ ✗

   Ignores the negative sign in the indexWorks out $9^{\frac{1}{2}} = 3$; the negative sign means you then take the reciprocal.

   D Option D: $-\frac{9}{2}$ ✗

   Multiplies the base by the indexWorks out $9 \times \left(-\frac{1}{2}\right)$; an index is not a multiplier.
5. ind-05 Present from here

   Work out $\left(\frac{2}{3}\right)^{-2}$

   A Option A: $\frac{4}{9}$

   B Option B: $\frac{9}{4}$

   C Option C: $-\frac{4}{9}$

   D Option D: $\frac{3}{2}$
   Answer and misconceptions

   A Option A: $\frac{4}{9}$ ✗

   Ignores the negative sign in the indexWorks out $\left(\frac{2}{3}\right)^2$; the negative index means the fraction must also be flipped.

   B Option B: $\frac{9}{4}$ ✓ Correct

   C Option C: $-\frac{4}{9}$ ✗

   Thinks a negative index makes the answer negativeSquares to get $\frac{4}{9}$ and makes it negative; a negative index means the reciprocal, not a negative number.

   D Option D: $\frac{3}{2}$ ✗

   Thinks a negative index only means take the reciprocalFlips the fraction but forgets the power of $2$: $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$.