Fractional Indices

Worked example

1. 1

   $9^{\frac{1}{2}}\times 9^{\frac{1}{2}}$ — Why is the power $\frac{1}{2}$ a square root? Multiply $9^{\frac{1}{2}}$ by itself and see.

2. 2

   $=9^{\frac{1}{2}+\frac{1}{2}}$ — Same number multiplied, so add the powers. That is the usual index law.

3. 3

   $=9^1=9$ — A half plus a half is 1, and $9^1$ is just 9.

4. 4

   $3\times 3=9$ — Which number times itself makes 9? It is 3.

5. 5

   $9^{\frac{1}{2}}=3=\sqrt{9}$ — Both $9^{\frac{1}{2}}$ and 3 make 9 when multiplied by themselves, so they are the same number: the square root of 9.

Worked example

1. 1

   $8^{\frac{2}{3}}$ — The question. The bottom 3 means cube root. The top 2 means square.

2. 2

   $\sqrt[3]{8}=2$ — Do the root first, because it makes the number small. 2 × 2 × 2 = 8, so the cube root of 8 is 2.

3. 3

   $2^2=4$ — Now the top 2: square what you have. 2 × 2 = 4.

4. 4

   $8^{\frac{2}{3}}=4$ — Done. Check the size: a power below 1 made 8 smaller.

Worked example

1. 1

   $32^{\frac{3}{5}}$ — Another one. The bottom 5 means fifth root. The top 3 means cube.

2. 2

   $\sqrt[5]{32}=2$ — Which number, used 5 times, multiplies to 32? 2 × 2 × 2 × 2 × 2 = 32, so it is 2.

3. 3

   $2^3=8$ — Now the top 3: cube it. 2 × 2 × 2 = 8.

4. 4

   $32^{\frac{3}{5}}=8$ — Done.

Worked example

1. 1

   $27^{-\frac{1}{3}}$ — The question. The minus says flip. The $\frac{1}{3}$ says cube root.

2. 2

   $=\frac{1}{27^{\frac{1}{3}}}$ — Deal with the minus first: write 1 over the same number, with the minus gone from the power.

3. 3

   $27^{\frac{1}{3}}=3$ — Now the cube root of 27: 3 × 3 × 3 = 27, so it is 3.

4. 4

   $=\frac{1}{3}$ — Put the 3 on the bottom. The answer is positive and less than 1.

Worked example

1. 1

   $16^{-\frac{3}{4}}$ — The question. The minus says flip, the bottom 4 a fourth root, the top 3 a cube.

2. 2

   $=\frac{1}{16^{\frac{3}{4}}}$ — Flip first: 1 over the number, and the minus is gone.

3. 3

   $\sqrt[4]{16}=2$ — The bottom 4: fourth root. 2 × 2 × 2 × 2 = 16, so it is 2.

4. 4

   $2^3=8$ — The top 3: cube it. 2 × 2 × 2 = 8, so $16^{\frac{3}{4}}=8$.

5. 5

   $=\frac{1}{8}$ — Put the 8 on the bottom of the flipped fraction.

Worked example

1. 1

   ${\left(\frac{8}{27}\right)}^{\frac{2}{3}}$ — The question. The power reaches the top and the bottom of the fraction.

2. 2

   $=\frac{8^{\frac{2}{3}}}{27^{\frac{2}{3}}}$ — Give the power to the top and to the bottom, then work each one out.

3. 3

   $=\frac{2^2}{3^2}$ — Bottom 3 means cube roots: the cube root of 8 is 2 and of 27 is 3.

4. 4

   $=\frac{4}{9}$ — Top 2 means square both: 2² = 4 and 3² = 9.

Worked example

1. 1

   ${\left(\frac{4}{9}\right)}^{-\frac{1}{2}}$ — The question. The minus says flip.

2. 2

   $={\left(\frac{9}{4}\right)}^{\frac{1}{2}}$ — Turn the whole fraction over. The minus is now used up, so the power is plain $\frac{1}{2}$.

3. 3

   $=\frac{\sqrt{9}}{\sqrt{4}}$ — The power $\frac{1}{2}$ means square root, on the top and on the bottom.

4. 4

   $=\frac{3}{2}$ — 3 × 3 = 9 and 2 × 2 = 4.

Worked example

1. 1

   $\frac{1}{\sqrt{7}}=7^n$ — The question. The right side is a power of 7, so make the left side one too.

2. 2

   $\sqrt{7}=7^{\frac{1}{2}}$ — A square root is the power one half.

3. 3

   $\frac{1}{7^{\frac{1}{2}}}=7^{-\frac{1}{2}}$ — 1 over a power is the same power with a minus.

4. 4

   $7^{-\frac{1}{2}}=7^n,n=-\frac{1}{2}$ — Both sides are now 7 to a power, so the powers must be equal.

Worked example

1. 1

   $\sqrt{2}\times 8=2^n$ — The question. Make every piece a power of 2.

2. 2

   $\sqrt{2}=2^{\frac{1}{2}}$ — A square root is the power one half.

3. 3

   $8=2^3$ — 2 × 2 × 2 = 8, so 8 is 2 cubed.

4. 4

   $2^{\frac{1}{2}}\times 2^3=2^{\frac{1}{2}+3}$ — Same base multiplied, so add the powers.

5. 5

   $\frac{1}{2}+\frac{6}{2}=\frac{7}{2}$ — Write 3 as $\frac{6}{2}$ so the bottoms match, then add the tops.

6. 6

   $n=\frac{7}{2}$ — Read off the power. 3.5 is also correct.

Worked example

1. 1

   $16^{1.5}$ — The question. The power 1.5 is a decimal.

2. 2

   $1.5=\frac{3}{2}$ — Write it as a fraction: 1.5 is three halves. Now the bottom 2 means square root, the top 3 means cube.

3. 3

   $\sqrt{16}=4$ — Root first: 4 × 4 = 16.

4. 4

   $4^3=64$ — Then cube: 4 × 4 × 4 = 64. So $16^{1.5}=64$.

Worked example

1. 1

   $8^{-\frac{2}{3}}\times {\left(5^4\right)}^{0.5}$ — Two parts multiplied. Work out the first part and the second part separately, then multiply.

2. 2

   $8^{-\frac{2}{3}}=\frac{1}{8^{\frac{2}{3}}}$ — First part. The minus flips it.

3. 3

   $\sqrt[3]{8}=2,2^2=4$ — Bottom 3: cube root of 8 is 2. Top 2: square it to get 4.

4. 4

   $8^{-\frac{2}{3}}=\frac{1}{4}$ — Put the 4 on the bottom. Write this value down: it earns a mark on its own.

5. 5

   $0.5=\frac{1}{2}$ — Second part. Change the decimal to a fraction: 0.5 is a half.

6. 6

   ${\left(5^4\right)}^{\frac{1}{2}}=5^{4\times \frac{1}{2}}$ — A power of a power: multiply the powers.

7. 7

   $5^2=25$ — 4 × ½ = 2, and 5² = 25. Write this value down too.

8. 8

   $\frac{1}{4}\times 25=\frac{25}{4}$ — Now multiply the two parts. $\frac{25}{4}$ is the same as 6.25.