0580 · 2.4Indices II

Core index laws

0:00Thirty-two raised to a power can give an answer smaller than thirty-two. In fact, raising it to the power of one fifth gives two. Fractional powers make this possible because they connect powers with roots. By the end of this video, you will understand that connection and use it to simplify expressions and solve equations. An index tells us the power to which a base is raised. In x³, x is the base and three is the index.

0:23For a positive whole number index, the power represents repeated multiplication. So x³ means three factors of x multiplied together, not 3 × x. When we multiply powers with the same base, we combine their factors. Multiplying x³ by x² gives five factors of x, so the result is x⁵. This explains why we add the indices. The base stays unchanged because every factor is still x. Division reverses that process by cancelling matching factors.

0:54Dividing x⁵ by x² leaves three factors of x. We therefore subtract the denominator’s index from the numerator’s index. This rule needs a base that is not zero, because division by zero is undefined. Raising a power to another power means multiplying the indices. For example, x², all cubed, contains three groups of two factors of x. That gives six factors altogether, producing x⁶. A power outside brackets also applies to every factor in a product, or to both parts of a fraction.

1:23These rules begin with counting factors, but they also work with negative and fractional indices wherever the expressions are defined. To understand those indices, we extend the same consistent pattern. Throughout our examples involving fractional powers of x, take x to be positive. This keeps roots and denominators defined without extra complications.

Zero, negative and fractional indices

1:42Look at descending powers of two. Each time the index decreases by one, the value is divided by two. Continuing past two to the power of one gives 2⁰ equal to one. The next step gives two to the power of negative one equal to 1/2. The division rule explains the zero index another way. Take a base that is not zero, cube it, then divide by itself, and the result is one.

2:06Subtracting the indices gives that same base to the power of zero. Therefore, any base other than zero, raised to the power of zero, equals one. A negative index means a reciprocal, not a negative answer. 2⁻³ = 1/2³, which is 1/8. Likewise, x⁻⁴ means 1/x⁴. The negative index changes where the power belongs in a fraction. Fractional indices describe roots because a root reverses a power. Squaring a to the power of one half multiplies its index by two, giving a to the power of one.

2:40So a to the power of one half is the square root of a. Similarly, a to the power of one third is the cube root of a. For a positive base, the denominator of a fractional index tells us which root to take. The numerator tells us which power to apply. Sixteen to the power of three over four means take the fourth root of sixteen, then cube the result.

2:59Root first is often easier because it keeps the numbers small. The fourth root of sixteen is two, since 2⁴ = 16. Cubing that result gives eight. For sixteen to the power of negative three over four, take the reciprocal of the positive power. The answer is therefore 1/8, combining the root rule with the negative index rule.

Simplifying fractional index expressions

3:21Cambridge questions use the command word simplify when asking you to write an equivalent expression in a simpler form. Separate the numerical coefficients from the powers of x before applying index rules. A coefficient is the number multiplying an algebraic part. This separation prevents you from accidentally treating a coefficient as an index. First, simplify three x to the power of negative four multiplied by two thirds x to the power of one half.

3:46Multiplying the coefficients, 3 × ⅔, gives two. The powers have the same base, so their indices are added. Write negative four as −8/2 to make that addition clear. −8/2 + 1/2 gives −7/2. Our expression becomes two x to the power of negative seven over two. Rewriting with a positive index gives two divided by x to the power of seven over two. These forms are equivalent, but follow any instruction about the form of your answer.

4:14Next, simplify two fifths x to the power of one half divided by two x to the power of negative two. Dividing the coefficients gives two fifths divided by two, which is one fifth. For the powers, subtract negative two from one half. Subtracting a negative becomes addition, giving ½ + 2. Write two as 4/2 before adding the fractions. 1/2 + 4/2 gives 5/2. The simplified expression is one fifth x to the power of five over two.

4:43We can write this as x to the power of five over two, all divided by five. Now simplify a fraction, all cubed. Its numerator is 2x⁵, and its denominator is three. The outer power applies to the whole fraction, including both coefficients. Cubing two gives eight, and raising x⁵ to the third power gives x¹⁵. Cubing the denominator gives twenty-seven, so the answer is eight x to the power of fifteen over twenty-seven.

Solving index equations and recap

5:11For equations involving indices, look for a common base on both sides. Different powers of the same positive base give different values, provided the base is not one. Equal values must therefore have equal indices. This lets us solve the supplied equations without logarithms. Solve 32ˣ = 2 by rewriting thirty-two as 2⁵. Raising that power to x gives 2⁵x. The right side is two to the power of one, so 5x = 1.

5:39Dividing both sides by five gives x equal to one fifth. Solve five to the power of x plus one equals twenty-five to the power of x by rewriting twenty-five as five squared. The right side becomes 5²x. Matching the indices gives x + 1 = 2x. Subtracting x from both sides gives 1 = x, so x = 1. Check the last answer by substituting one into the original equation.

6:05The left side becomes 5², which equals twenty-five. On the right, twenty-five to the power of one also gives twenty-five. Matching results confirm that the solution satisfies the original equation. Common mistakes include adding indices when the expressions themselves are being added. For example, x² + x³ cannot become x⁵. The index addition rule applies to multiplication of powers with the same base. It counts factors, not separate terms joined by addition.

6:33Another mistake is applying an outside power to only part of a bracket. Our cubed fraction required cubing two and three as well as multiplying the index of x by three. Also, keep negative signs separate from negative indices. −2² inside brackets gives four, whereas 2⁻² gives 1/4. These methods must work without a calculator on Paper two, and they are also useful on calculator Paper four. Keep fractions exact in these questions, rather than replacing them with rounded decimals.

7:02For numerical answers that require rounding, follow the stated accuracy or the usual three significant figures. Never round intermediate calculations. The central idea is that index rules preserve a consistent pattern of multiplication and division. Multiplying powers adds indices, dividing subtracts them, and raising a power multiplies them. A zero index gives one for any base other than zero, a negative index gives a reciprocal, and a fractional index introduces a root.

7:30Using these connections makes unfamiliar expressions understandable rather than mysterious.