Negative and Fractional Indices

A negative index means take the reciprocal of the corresponding positive power.

1. The negative index gives 5^{-2}=\frac{1}{5^2}.
2. Since 5^2=25, the value is \frac{1}{25}.

An index of one half means take the positive square root.

1. The denominator 2 in the index means square root.
2. The positive square root of 49 is 7, so 49^{\frac{1}{2}}=7.

Think of the number which, multiplied by itself three times, gives the base.

1. An index of \frac{1}{3} means cube root.
2. Since 4^3=64, the value is 4.

Take the fourth root first, then cube it.

1. In 16^{\frac{3}{4}}, the denominator 4 means fourth root and the numerator 3 means cube.
2. The fourth root of 16 is 2, because 2^4=16.
3. Cube it: 2^3=8. So 16^{\frac{3}{4}}=8.

Find the positive fractional power first, then take its reciprocal.

1. Deal with the fraction first: 1000^{\frac{2}{3}} means the cube root of 1000, squared.
2. The cube root of 1000 is 10, and 10^2=100, so 1000^{\frac{2}{3}}=100.
3. The negative sign in the index means take the reciprocal: 1000^{-\frac{2}{3}}=\frac{1}{100}. It is not -100: a negative index never makes the value negative.

Express the denominator as a power of five, then use the reciprocal rule.

1. Since 125=5^3, the fraction is \frac{1}{5^3}.
2. The reciprocal is written with a negative index: 5^{-3}.
3. Therefore n=-3.

Write the base as a power of three and compare the exponents.

1. Write 81=3^4. The equation becomes (3^4)^n=3^1.
2. Multiply the indices on the left to get 3^{4n}=3^1.
3. Equal powers of 3 have equal exponents, so 4n=1 and n=\frac{1}{4}.

The square root applies to both numerator and denominator; the negative index then reverses the fraction.

1. The power \frac{1}{2} means square root, applied to the numerator and the denominator: \left(\frac{9}{16}\right)^{\frac{1}{2}}=\frac{3}{4}.
2. The negative sign in the index means take the reciprocal, which turns the fraction upside down.
3. So \left(\frac{9}{16}\right)^{-\frac{1}{2}}=\frac{4}{3}.

Evaluate each fractional power separately before multiplying.

1. Work out each power separately. 32^{\frac{2}{5}} is the fifth root of 32, squared: the fifth root is 2 (as 2^5=32), and 2^2=4.
2. 8^{\frac{1}{3}} is the cube root of 8, which is 2. The negative index means reciprocal, so 8^{-\frac{1}{3}}=\frac{1}{2}.
3. Multiply: 4\times\frac{1}{2}=2.

Write \sqrt{5} and 25 separately as powers of 5.

1. Write each part as a power of 5: a square root is a power of \frac{1}{2}, so \sqrt{5}=5^{\frac{1}{2}}, and 25=5^2.
2. So \frac{\sqrt{5}}{25}=5^{\frac{1}{2}}\div5^2.
3. When dividing powers of the same number, subtract the indices: \frac{1}{2}-2=-\frac{3}{2}.
4. So \frac{\sqrt{5}}{25}=5^{-\frac{3}{2}} and k=-\frac{3}{2}.

Write both sides as powers of two.

1. Since 16=2^4, the left side is 2^{4x}.
2. The right side is 2^{-3}, so 4x=-3.
3. Dividing by 4 gives x=-\frac{3}{4}.

Write 4^n and 8 as powers of 2 first.

1. Write everything as a power of 2: 4=2^2, so 4^n=(2^2)^n=2^{2n}, and 8=2^3.
2. Multiplying powers of 2 adds the indices: 2^{2n}\times2^3=2^{2n+3}.
3. Dividing subtracts the indices: 2^{2n+3}\div2^{n+1}=2^{(2n+3)-(n+1)}.
4. Simplify the index: 2n+3-n-1=n+2. So k=n+2.

The power outside the bracket applies to both the number and the power of the letter.

1. Take the cube root of the numerical factor: 27^{\frac{1}{3}}=3.
2. Multiply the indices on the letter: (a^6)^{\frac{1}{3}}=a^2.
3. The simplified expression is 3a^2.

Find the square root of the bracket, then take its reciprocal.

1. The positive square root of 16x^8 is 4x^4.
2. The negative index requires the reciprocal of this result.
3. The expression simplifies to \frac{1}{4x^4}.

Deal with the numbers, the powers of x and the powers of y separately, subtracting indices when dividing.

1. Divide the numbers: 6\div2=3.
2. Divide the powers of x by subtracting indices: x^{-2}\div x^3=x^{-2-3}=x^{-5}.
3. Divide the powers of y by subtracting indices: y^3\div y^{-1}=y^{3-(-1)}=y^4.
4. So the expression is 3x^{-5}y^4. Since x^{-5}=\frac{1}{x^5}, this is \frac{3y^4}{x^5}.

For each bracket, take the root, then the power, then the reciprocal.

1. First bracket: the cube root of \frac{125}{8} is \frac{5}{2}. Squaring gives \frac{25}{4}, and the negative index turns it upside down: \frac{4}{25}.
2. Second bracket: the square root of \frac{4}{25} is \frac{2}{5}, and the negative index turns it upside down: \frac{5}{2}.
3. Multiply: \frac{4}{25}\times\frac{5}{2}=\frac{20}{50}.
4. Simplify by dividing by 10: \frac{2}{5}.

Apply the fractional power to the numerator before using the division rule for indices.

1. A power outside a bracket applies to every factor inside it, so find 32^{\frac{3}{5}} and (a^{10})^{\frac{3}{5}} separately.
2. The fifth root of 32 is 2, so 32^{\frac{3}{5}}=2^3=8.
3. For a power of a power, multiply the indices: (a^{10})^{\frac{3}{5}}=a^{10\times\frac{3}{5}}=a^6.
4. The fraction is now \frac{8a^6}{4a^2}. Divide the numbers, 8\div4=2, and subtract the indices, 6-2=4, to get 2a^4.

Write both bases as powers of three before equating the exponents.

1. Use 9=3^2 and 27=3^3. The equation becomes 3^{2(x+1)}=3^{3(2x-1)}.
2. Equate exponents: 2x+2=6x-3.
3. Rearrange to get 5=4x.
4. Therefore x=\frac{5}{4}.

Use powers of two to find the first exponent and powers of three to find the second.

1. Write the first equation as 2^{4p}=2^3, giving p=\frac{3}{4}.
2. Write the second equation as 3^{3q}=3^{-2}, giving q=-\frac{2}{3}.
3. Add using denominator 12: p+q=\frac{9}{12}-\frac{8}{12}.
4. The answer is \frac{1}{12}.

Write 8, x^2 and \sqrt{y} each as a power of 2.

1. Write each part as a power of 2: 8=2^3, and x^2=(2^a)^2=2^{2a} (power of a power: multiply the indices).
2. A square root is a power of \frac{1}{2}, so \sqrt{y}=(2^b)^{\frac{1}{2}}=2^{\frac{b}{2}}.
3. Multiply the top by adding indices: 2^3\times2^{2a}=2^{2a+3}.
4. Divide by subtracting indices: \frac{2^{2a+3}}{2^{\frac{b}{2}}}=2^{2a+3-\frac{b}{2}}.
5. So k=2a-\frac{b}{2}+3.