Apply algebra is essential for solving quantitative problems across all professional fields in Kenya, from calculating financial growth in banks and SACCOs to determining dosage in healthcare and optimizing resource use in agriculture. Mastery of indices algebra allows professionals to simplify expressions, model growth or decay, and handle complex calculations efficiently. This chapter introduces indices, their laws, and practical applications illustrated through examples from diverse sectors such as county hospitals, retail businesses, and universities.

1.1 Indices algebra

Indices algebra deals with the rules and operations involving powers or exponents. Understanding indices enables professionals to manipulate expressions involving repeated multiplication and roots, which frequently appear in calculations like compound interest, population growth, and scientific measurements.

1.1.1 Power zero

The zero power law states that any nonzero number raised to the power of zero equals one. This rule helps simplify expressions and solve equations where variables are raised to zero power.

$$a^{0} = 1 \quad \text{for} \quad a \neq 0$$

Worked Examples

Example 1: A county referral hospital has a medication dosage formula involving \(x^0\), where \(x = 5\) mg. Calculate \(5^0\).

Given: \(x = 5\)

$$x^{0} = 1$$

Answer: 1

Example 2: A university's research data includes the expression \( (3y)^0 \), where \(y = 10\). Find the value.

Given: \(y = 10\)

$$(3y)^0 = 1$$

Answer: 1

Example 3: In a retail business, the turnover rate is modeled by \(k^0\) where \(k = 0.75\). Calculate \(k^0\).

Given: \(k = 0.75\)

$$k^{0} = 1$$

Answer: 1

1.1.2 Negative powers

Negative powers represent the reciprocal of the base raised to the corresponding positive power. This concept is vital for handling division and inverse relationships in calculations such as depreciation rates and concentration dilutions.

$$a^{-n} = \frac{1}{a^{n}} \quad \text{for} \quad a \neq 0, \, n > 0$$

Worked Examples

Example 1: A SACCO calculates loan repayment inversely proportional to \(x^{-2}\), with \(x = 4\). Find the value.

Given: \(x = 4\)

$$x^{-2} = \frac{1}{x^{2}}$$

$$= \frac{1}{4^{2}}$$

$$= \frac{1}{16}$$

Answer: \frac{1}{16} = 0.0625$$

Example 2: A farm's pesticide concentration is modeled as \(C = 5t^{-3}\), where \(t = 2\) hours after application. Calculate \(C\).

Given: \(t = 2\), constant 5

$$C = 5 \times t^{-3} = 5 \times \frac{1}{t^{3}}$$

$$= 5 \times \frac{1}{2^{3}}$$

$$= 5 \times \frac{1}{8}$$

$$= \frac{5}{8} = 0.625$$

Answer: 0.625

Example 3: A county government office models pollutant reduction as \(P = 100x^{-1.5}\), where \(x = 4\). Find \(P\).

Given: \(x = 4\), constant 100

$$P = 100 \times x^{-1.5} = 100 \times \frac{1}{x^{1.5}}$$

$$= 100 \times \frac{1}{4^{1.5}}$$

Calculate \(4^{1.5}\):

$$4^{1.5} = 4^{1 + 0.5} = 4^{1} \times 4^{0.5} = 4 \times 2 = 8$$

$$P = 100 \times \frac{1}{8} = 12.5$$

Answer: 12.5

1.1.3 Fractional powers

Fractional powers denote roots, with the numerator as the power and denominator as the root degree. This rule applies in fields like agriculture for growth rates, finance for interest calculations, and healthcare for dosage adjustments.

$$a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m} \quad \text{for} \quad a > 0$$

Worked Examples

Example 1: A university research involves calculating \(\sqrt[3]{27}\) expressed as \(27^{\frac{1}{3}}\).

Given: \(a = 27\)

$$27^{\frac{1}{3}} = \sqrt[3]{27}$$

Calculate cube root:

$$\sqrt[3]{27} = 3$$

Answer: 3

Example 2: A hotel calculates compound interest using \(A = P \times r^{\frac{1}{2}}\), with \(r = 1.44\). Find \(r^{\frac{1}{2}}\).

Given: \(r = 1.44\)

$$r^{\frac{1}{2}} = \sqrt{1.44}$$

Calculate the square root:

$$\sqrt{1.44} = 1.2$$

Answer: 1.2

Example 3: A county hospital measures drug concentration as \(C = 50^{\frac{3}{2}}\). Calculate \(C\).

Given: \(a = 50\)

$$50^{\frac{3}{2}} = (50^{\frac{1}{2}})^{3} = (\sqrt{50})^{3}$$

Calculate \(\sqrt{50}\):

$$\sqrt{50} = 7.071$$

Then cube:

$$7.071^{3} = 7.071 \times 7.071 \times 7.071$$

Step 1:

$$7.071 \times 7.071 = 50$$ (approximate)

Step 2:

$$50 \times 7.071 = 353.55$$

Answer: 353.55

1.1.4 Laws of indices

The laws of indices simplify expressions involving multiplication, division, and powers of powers. Mastery of these laws is critical for professionals in all sectors to efficiently compute and simplify formulas.

The key laws are:

• Product law: \(a^{m} \times a^{n} = a^{m+n}\)
• Quotient law: \(\frac{a^{m}}{a^{n}} = a^{m-n}\)
• Power of a power: \((a^{m})^{n} = a^{m \times n}\)
• Power of a product: \((ab)^{n} = a^{n} \times b^{n}\)
• Power of a quotient: \((\frac{a}{b})^{n} = \frac{a^{n}}{b^{n}}\)

Worked Examples

Example 1: A retail business calculates revenue growth as \(R = 2^{3} \times 2^{4}\). Find \(R\).

Given: \(2^{3} \times 2^{4}\)

Apply product law:

$$2^{3+4} = 2^{7}$$

Calculate:

$$2^{7} = 128$$

Answer: 128

Example 2: A SACCO models loan interest ratio as \(\frac{5^{6}}{5^{2}}\). Simplify.

Given: \(\frac{5^{6}}{5^{2}}\)

Apply quotient law:

$$5^{6-2} = 5^{4}$$

Calculate:

$$5^{4} = 625$$

Answer: 625

Example 3: A county government office calculates pollutant increase as \((3^{2})^{4}\). Simplify.

Given: \((3^{2})^{4}\)

Apply power of a power law:

$$3^{2 \times 4} = 3^{8}$$

Calculate:

$$3^{8} = 6561$$

Answer: 6561

Example 4: A farm calculates fertilizer amount as \((2 \times 3)^{3}\). Simplify.

Given: \((2 \times 3)^{3}\)

Apply power of a product:

$$2^{3} \times 3^{3} = 8 \times 27 = 216$$

Answer: 216

Example 5: A university calculates ratio \((\frac{4}{2})^{3}\). Simplify.

Given: \((\frac{4}{2})^{3}\)

Apply power of a quotient:

$$\frac{4^{3}}{2^{3}} = \frac{64}{8} = 8$$

Answer: 8

Chapter Summary

This chapter covered essential algebraic concepts starting with indices algebra, including the rules for zero, negative, and fractional powers, as well as the fundamental laws governing addition, subtraction, multiplication, and division of indices. The importance of BODMAS in correctly evaluating mathematical expressions was emphasized to ensure accurate calculations. The chapter then explored roots, detailing square roots, cube roots, and nth roots to solve various algebraic problems. Logarithms were introduced, with a focus on their key laws such as the product, quotient, and power laws that simplify complex expressions. Practical use of the scientific calculator was demonstrated, highlighting functions like power on/off, mode selection including degrees, radians, gradients, and the use of clear, save, and shift keys. Methods for solving simultaneous equations with up to three variables were examined through elimination, substitution, reduction, and graphical approaches. Finally, quadratic equations were addressed using factorization, the quadratic formula, completing the square, and graphical methods to find solutions efficiently.