0580 · 1.7Indices I

Index laws with positive powers

0:002³ = 8, but 2⁰ = 1. A negative power gives a fraction, not a negative number. These results follow one connected pattern. By the end, you will understand how powers, roots and reciprocals fit together, and why the rules of indices work. You need multiplication, division, fractions and a basic understanding of roots. A square root reverses squaring, while a cube root reverses cubing. A reciprocal is one divided by a number, so the reciprocal of eight is 1/8.

0:30We will use these familiar operations to make unfamiliar powers understandable. In 3⁴, three is the base and four is the index. Another word for index is exponent. The plural of index is indices. A positive whole number index tells us how many copies of the base are multiplied together. Here, four copies of three give eighty-one, rather than 3 × 4. Three to the power of one is simply three.

0:56Squared means to the power of two, and cubed means to the power of three. Repeated multiplication explains positive whole number powers. To understand other indices, we extend that idea while keeping the rules consistent. Consider 2³ × 2². The first power contains three twos, and the second contains two more. Altogether, five twos are multiplied, giving 2⁵, which equals thirty-two. This explains why multiplying powers with the same base means adding their indices.

1:25Division removes matching factors instead of joining them. 2⁵ ÷ 2³ leaves two factors of two after cancellation. The result is 2², which equals four. Therefore, when dividing powers with the same nonzero base, subtract the second index from the first. When brackets are raised to an index, the outer index applies to everything inside the brackets. 2³, all squared, means 2³ multiplied by itself. Adding three and three gives six, so the answer is 2⁶, or sixty-four.

1:57So for a power of a power, we multiply the indices. Brackets can also contain a product or a fraction. Take 3 × 2, all squared. Each factor is squared, giving 9 × 4, which equals thirty-six. With 3/2, all squared, the numerator and the denominator are both squared, giving 9/4. Powers distribute over multiplication and division, but not over addition. For example, 3 + 2, all squared, gives 5², which equals twenty-five.

2:28Squaring the two terms separately and adding gives thirteen instead. These are different calculations because addition does not join repeated factors. Always check the operation before choosing an index rule.

Zero and negative indices

2:39Follow the powers of two downwards, starting with 2³, which equals eight. Reducing the index by one divides the value by two. The next values are four, then two, then one, then 1/2. Keeping the same pattern explains both zero indices and negative indices. Every nonzero number to the power of 0 = 1. We can justify this by dividing 5³ by itself, which gives one. Subtracting the indices gives 5⁰, so that must also equal one.

3:07The nonzero condition matters because division by zero is undefined. A negative index means take the reciprocal of the corresponding positive power. 2⁻³ therefore equals 1 ÷ 2³. Since 2³ = 8, the answer is 1/8. The negative index moves the power into the denominator. It does not make the answer negative. Our first exam style calculation is to find the value of 7⁻². Rewrite the negative power as 1/7².

3:397² = 49, giving the exact answer 1/49. This method needs no calculator and avoids confusing a negative index with a negative value. Next, find the value of 2⁻³ × 2⁴. The bases match, so add negative three and four. Their sum is one, giving two to the power of one. The answer is two, showing that the multiplication rule still works with negative indices. Now find the value of 2³ ÷ 2⁴.

4:09Subtract four from three because this is division with matching bases. The result is two to the power of negative one. Rewriting that as a reciprocal gives 1/2. This matches 8 ÷ 16.

Fractional and negative fractional indices

4:22A fractional index represents a root because the power of a power rule must remain consistent. Squaring six to the power of one over two multiplies its index by two. The resulting index is one, so the squared value is six. Therefore, six to the power of one over two is the square root of six. For positive bases, an index of one over a whole number means take the corresponding root.

4:45Sixteen to the power of one over four means the fourth root of sixteen. Its value is two because 2⁴ = 16. The denominator of a fractional index tells us which root to take. To find the value of eighty-one to the power of one over two, first recognise the index as a square root. √81 is nine because 9² = 81. Therefore, the answer is nine. The square root symbol means the positive value only, so we do not add a negative answer here.

5:16When the numerator is not one, it tells us another power to apply. Eight to the power of two over three means take the cube root, then square the result. The cube root of eight is two, and 2² is four. Taking the root first usually keeps the numbers smaller. For a complete Extended example, find the value of eight to the power of negative two over three. The negative index gives one divided by eight to the power of two over three.

5:40The denominator becomes the cube root of eight, all squared. Replacing that root with two gives 1/2², which equals 1/4. That single index holds three instructions. The bottom number, three, means take the cube root. Next, the top number, two, squares that result. Finally, the negative sign turns it into a reciprocal, and every step stays exact.

Common mistakes and exam technique

6:04One common mistake is to multiply the indices when two powers are multiplied together. The reverse mistake is to add the indices for a power of a power. Think about factors to choose correctly. Multiplication joins two groups of factors, so their sizes add. A power of a power repeats the whole group, so the sizes multiply. Another mistake is using these rules when the bases differ. 2³ × 3² cannot become 6⁵.

6:32Also, a negative base differs from a negative index. Negative two, all squared, equals four, whereas 2⁻² = 1/4. An examination may ask you to find the value of a numerical expression or simplify it using index rules. These exact methods must work without a calculator on Paper two, and remain useful on calculator Paper four. Keep fractions and roots exact when an exact answer is required. For decimal answers, follow the stated accuracy; otherwise, normally use three significant figures without rounding intermediate values.

7:05Indices form one connected system rather than a collection of unrelated tricks. Positive whole number powers count repeated factors, while zero powers and negative powers continue the division pattern. Fractional powers reverse powers through roots. Identify the base and operation first, then use the rule that matches the structure of the expression.