Exponents and Radicals
Introduction
Exponents (indices) are a short way to write repeated multiplication, while radicals (roots) reverse the process. In the Kenyan Senior School CBC (KICD) Mathematics curriculum, they are used in science, finance (compound interest), and population growth.
Exponent notation
a^n means a multiplied by itself n times. Here a is the base and n is the exponent (index or power). For example, 3^4 = 3 x 3 x 3 x 3 = 81.
Laws of exponents
- a^m x a^n = a^(m+n)
- a^m / a^n = a^(m-n)
- (a^m)^n = a^(mn)
- (ab)^n = a^n b^n
- a^0 = 1 (for a not zero)
- a^-n = 1 / a^n
- a^(1/n) = nth root of a; a^(m/n) = (nth root of a)^m
Worked example 1
Simplify (x^3 y^2)^2 / (x y). Answer: (x^6 y^4) / (x y) = x^5 y^3.
Worked example 2
Solve 2^(x+1) = 32. Answer: 32 = 2^5, so x + 1 = 5 and x = 4.
Negative and fractional indices
- 4^-1 = 1/4
- 16^(1/2) = 4
- 8^(-1/3) = 1/(8^(1/3)) = 1/2
Radicals and surds
A surd is a root that is irrational, such as sqrt(2), sqrt(5) or cube root of 7. Note sqrt(9) = 3 is not a surd.
Rules:
- sqrt(a) x sqrt(b) = sqrt(ab)
- sqrt(a) / sqrt(b) = sqrt(a/b)
- sqrt(a) + sqrt(a) = 2 sqrt(a)
Simplifying surds
Find the largest square factor. sqrt(72) = sqrt(36 x 2) = 6 sqrt(2).
Rationalising the denominator
To remove a surd from the denominator:
- 6 / sqrt(3) = 6 sqrt(3) / 3 = 2 sqrt(3).
- 1 / (2 + sqrt(3)): multiply top and bottom by the conjugate (2 - sqrt(3)). The denominator becomes 4 - 3 = 1, so the result is 2 - sqrt(3).
Worked example 3
Simplify 3 sqrt(12) - sqrt(27). Answer: 3 x 2 sqrt(3) - 3 sqrt(3) = 6 sqrt(3) - 3 sqrt(3) = 3 sqrt(3).
Standard form
Large and small numbers are written as A x 10^n with 1 <= A < 10. For example, 45,000 = 4.5 x 10^4 and 0.0032 = 3.2 x 10^-3.
Try it
- Evaluate 3^-2 x 3^4.
- Simplify sqrt(98) and sqrt(75).
- Rationalise 10 / sqrt(5).
- Solve 3^(2x) = 81.
- Write 6,300,000 in standard form.