Fractional Indices with Letters

Worked example

1. 1

   $8^{\frac{2}{3}}$ — The question. The power is a fraction, so it is a root and a power together.

2. 2

   $\sqrt[3]{8}=2$ — The bottom 3 means cube root. 2 × 2 × 2 = 8, so the cube root of 8 is 2. Doing the root first keeps the number small.

3. 3

   $2^2=4$ — The top 2 means square what you have: 2 × 2 = 4. So $8^{\frac{2}{3}}=4$.

Worked example

1. 1

   $(27p^6{q}^{12}{)}^{\frac{2}{3}}$ — The question. Three parts are inside the bracket: 27, $p^6$ and $q^{12}$.

2. 2

   $27^{\frac{2}{3}}\times (p^6{)}^{\frac{2}{3}}\times (q^{12}{)}^{\frac{2}{3}}$ — Give the power $\frac{2}{3}$ to each part, because the bracket is one product.

3. 3

   $\sqrt[3]{27}=3$ — The number first. The bottom 3 means cube root: 3 × 3 × 3 = 27.

4. 4

   $3^2=9$ — The top 2 means square it: 3² = 9.

5. 5

   $p^{6\times \frac{2}{3}}=p^4$ — A letter: multiply the powers. 6 × ⅔ = 4, because a third of 6 is 2 and two of those is 4.

6. 6

   $q^{12\times \frac{2}{3}}=q^8$ — The same for q: a third of 12 is 4, and 2 × 4 = 8.

7. 7

   $9p^4{q}^8$ — Write the three parts next to each other. This is the answer.

Worked example

1. 1

   ${\left(\frac{4x^6}{9y^2}\right)}^{-\frac{3}{2}}$ — The question. The power has a minus, so flip first.

2. 2

   ${\left(\frac{9y^2}{4x^6}\right)}^{\frac{3}{2}}$ — Flip the fraction and drop the minus. The power is now $\frac{3}{2}$.

3. 3

   $9^{\frac{3}{2}}=(\sqrt{9}{)}^3=3^3=27$ — The top number. The bottom 2 means square root, $\sqrt{9}=3$, then the top 3 means cube: 27.

4. 4

   $4^{\frac{3}{2}}=(\sqrt{4}{)}^3=2^3=8$ — The bottom number, the same way: $\sqrt{4}=2$, and 2³ = 8.

5. 5

   $y^{2\times \frac{3}{2}}=y^3$ — A letter: multiply the powers. 2 × 3/2 = 3.

6. 6

   $x^{6\times \frac{3}{2}}=x^9$ — The other letter: 6 × 3/2 = 9, because half of 6 is 3 and 3 × 3 = 9.

7. 7

   $\frac{27y^3}{8x^9}$ — Put each part back where it was after the flip: y on top, x underneath.

Worked example

1. 1

   ${\left(\frac{x^2{y}^5}{x^{8}y}\right)}^{-\frac{1}{2}}$ — The question. The same letters are on the top and the bottom, so tidy inside first.

2. 2

   $x^{2-8}=x^{-6}$ — Dividing means subtract the powers: top minus bottom, 2 − 8 = −6.

3. 3

   $y^{5-1}=y^4$ — The same for y. A letter on its own has power 1, so 5 − 1 = 4.

4. 4

   ${\left(x^{-6}{y}^4\right)}^{-\frac{1}{2}}$ — The bracket is now one simple product. Use the outside power next.

5. 5

   $(-6)\times \left(-\frac{1}{2}\right)=+3$ — The x power: minus times minus is plus, so it is +3.

6. 6

   $4\times \left(-\frac{1}{2}\right)=-2$ — The y power: plus times minus is minus, so it is −2.

7. 7

   $x^3{y}^{-2}$ — Put the new powers on the letters.

8. 8

   $=\frac{x^3}{y^2}$ — A negative power means 1 over, so $y^{-2}$ moves underneath. Both answers are correct unless the question asks for a form.

Worked example

1. 1

   ${\left(\frac{\sqrt[3]{y}}{2}\right)}^{-1}$ — The question: write it in the form $c{y}^n$.

2. 2

   $=\frac{2}{\sqrt[3]{y}}$ — The power −1 flips the fraction: the 2 goes on top.

3. 3

   $=\frac{2}{y^{\frac{1}{3}}}$ — A cube root is the power ⅓.

4. 4

   $=2y^{-\frac{1}{3}}$ — 1 over a power is the same power with a minus. So c = 2 and n = −⅓.

Worked example

1. 1

   $\frac{\sqrt[3]{x}}{x^2}$ — The question: write it as one power of x.

2. 2

   $=\frac{x^{\frac{1}{3}}}{x^2}$ — Change the root to a power first, so both parts are powers of x.

3. 3

   $=x^{\frac{1}{3}-2}$ — Dividing means subtract the powers: top minus bottom.

4. 4

   $\frac{1}{3}-\frac{6}{3}$ — To subtract, give both the same bottom: 2 = 6/3.

5. 5

   $=x^{-\frac{5}{3}}$ — 1 − 6 = −5, so the power is −5/3. A minus answer is fine here: it means 1 over.