Year 10 Extended · Unit 1 Number sets, surds, indices and bounds · Lesson 5 of 6

What could it mean to raise a number to the power one half?

The laws of indices were written for whole-number powers. This lesson asks what a fractional power would have to mean for those laws to keep working, and tests the answer on growth over part of a time period.

Number and AlgebraGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: write x⁵ × x³ as a single power.
    Answerx⁸. Add the exponents: 5 + 3 = 8.
  2. From Year 9: evaluate 2−3.
    Answer1/8. A negative exponent means a reciprocal: 1 ÷ 2³.
  3. What is √(16 × 9)?
    Answer12. √144 = 12, or 4 × 3.
Words used in this lesson
index
the small raised number that says how many times to multiply. For example: the 5 in 2⁵.
indices
more than one index. For example: the 3 and the 7 in 4³ and 4⁷.
root
the number that, multiplied by itself n times, gives another number. For example: the cube root of 27.
reciprocal
1 divided by a number. For example: the reciprocal of 4 is 1/4.
base
the number that is being raised to a power. For example: the 10 in 10⁻².
equivalent
different in form but equal in value. For example: √x and x1/2.

Splitting a step in half

A bacterial culture grows by the same factor in every hour, and over five hours it multiplies by 32. The growth over any stretch of time is a power of that single hourly factor, so a part of the five hours can be written with a fractional exponent: the growth over three hours is 323/5.

Predict first: what is 323/5: 32 × 3/5, 8 or 96/5? And what does the hourly factor, 321/5, have to be, if five equal multiplications make 32? Use the figures, then check.

1
3
64¹⁄³ = 4value
0.33exponent

64¹⁄³: the cube root of 64 is 4 (4³ = 64), then 4 to the power 1 gives 4.

Show me the steps
  1. 64 to the power 1/3: split the step from exponent 0 to 1 into 3 equal multiplications. One factor r repeated three times makes 64, so r = 4, the cube root.
  2. Set m = 2 and n = 3: the cube root of 64 is 4 (4³ = 64), and 4 to the power 2 is 16, so 64 to the power 2/3 is 16.
  3. Set m = 3 and n = 4: the 4th root of 64 is not a whole number (it is 2.83…), and the value is about 22.627.
  4. Choose base 81 with m = −2 and n = 4: the 4th root of 81 is 3, and 3 to the power −2 is 1/9. A negative exponent takes the reciprocal.
2
4same value
64biggest number on the power-first route

8²⁄³: root first, the cube root of 8 is 2 and 2² = 4; power first, 8² = 64 and the cube root of that is 4. Same value, smaller numbers.

Show me the steps
  1. 8 to the power 2/3: root first, the cube root of 8 is 2 and 2² = 4. Power first, 8² = 64 and the cube root of 64 is 4. Same value.
  2. Choose 1000 with m = 4: root first, the cube root of 1000 is 10 and 10⁴ = 10 000. Power first, 1000⁴ = 1 000 000 000 000 and then the cube root. The same value, but the numbers are huge.
  3. Choose 32 with m = −2: root first, the 5th root of 32 is 2 and 2 to the power −2 is 1/4. A negative exponent is a reciprocal, whichever route you take.
  4. Set m to 0: every base gives 1. Zero steps of the repeated multiplication leave the starting 1.

For x > 0, x1/n is the nth root of x, the number that makes x when multiplied by itself n times. Then xm/n = (n√x)m = n√(xm): a root and a power, in either order, and the root first usually keeps the numbers small.

The laws of indices keep working: xa × xb = xa + b, (xa)b = xab, and x−a = 1/xa. That is why the definition has to be what it is: (x1/n)n = x1.

The denominator is the root; the numerator is the power.

Two common mistakes. Reading x3/2 as x × 3/2, or as the cube root of x squared: the 3 is a power and the 2 is a root. And forgetting that a negative exponent does not make the answer negative: 4−1/2 is 1/2, a reciprocal, not −2.

Worked examples

1. Evaluate 2563/4.

  1. The denominator 4 is the 4th root: 4√256 = 4, because 4⁴ = 256.
  2. The numerator 3 is the power: 4³ = 64.

2. Simplify (27x⁶)−2/3.

  1. Apply the power to each factor: 27−2/3 × (x⁶)−2/3.
  2. 272/3 = (³√27)² = 3² = 9, so 27−2/3 = 1/9. And (x⁶)−2/3 = x−4 = 1/x⁴.
  3. The result is 1/(9x⁴).

3. Solve 9x = 27.

  1. Write both sides as powers of 3: 9 = 3² and 27 = 3³, so 32x = 3³.
  2. The bases are equal, so the exponents are: 2x = 3 and x = 3/2.

Your go. The first step is given. Evaluate (4/9)−3/2.

  1. A negative exponent is a reciprocal: (4/9)−3/2 = (9/4)3/2.
  2. Take the square root, then cube it.
Check the answer

√(9/4) = 3/2, and (3/2)³ = 27/8.

With a GDC (optional)

A GDC has a power key, usually marked ^. A fractional exponent must be in brackets: to enter 10 to the power 3/2, type 10 ^ (3 ÷ 2). Without the brackets the GDC works out 10³ first and then divides by 2, giving 500, which is a different number. A negative exponent also goes in brackets.

A GDC shows a decimal, so a written answer should still be in exact form (a whole number or a fraction) when the root is exact. Use the GDC to check, not to think: everything here can be done by hand.

Check before you go on

1. For x > 0, which is the same as x2/3?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Evaluate 493/2.

3. Evaluate 625−3/4 as a decimal.

Level 2 · Building

4. (125x⁹)2/3 = a xb. What is a?

5. x1/2 × x1/3 = xp/6. What is p?

6. For x > 0, which is equal to ³√(x²) × √x?

Level 3 · Stretch

7. A yeast colony in a laboratory grows by a factor of 4 every hour. After how many hours has it grown by a factor of 32? Solve 4x = 32.

8. x3/2 = 27 and x > 0. Find x.

Something to think about

Why does the definition need x > 0? Try (−8) to the power 1/3 as the cube root: −2. Now write 1/3 as 2/6 and use the 6th root of (−8)². What goes wrong, and what does it tell you about the rule xm/n for negative x?

One way to think about it

(−8)1/3 = −2, but 2/6 is the same fraction, and (−8)2/6 is the 6th root of 64, which is +2. The same exponent gives two different answers, so the rule would not be well defined. Restricting to x > 0 keeps a power's value independent of how the fraction is written.

Exit ticket

9. For x > 0, write 1/³√(x⁴) as a single power of x.