Indices, Logarithms and Surds

Question 1

What are the laws of indices, and how do you apply them in calculations?

Answer 1

1. Multiplication: 𝑎ᵐ × 𝑎ⁿ = 𝑎ᵐ⁺ⁿ
2. Division: 𝑎ᵐ ÷ 𝑎ⁿ = 𝑎ᵐ⁻ⁿ
3. Power of a power: (𝑎ᵐ)ⁿ = 𝑎ᵐⁿ
4. Zero exponent: 𝑎⁰ = 1 (where 𝑎 ≠ 0)
5. Negative exponent: 𝑎⁻ⁿ = 1 / 𝑎ⁿ
6. Fractional exponent: 𝑎 ᵐ/ⁿ = ⁿ√𝑎ᵐ

Question 2

How do you convert numbers to standard form?

Answer 2

1. Identify the decimal point in the number.
2. Move the decimal point so that only one non-zero digit remains on the left.
3. Count the number of places the decimal moved; this becomes the exponent of 10.
4. Write the number as a product of the adjusted number and
   10 raised to the power of the exponent.

Question 2

How do you establish the relationship between indices and logarithms in solving problems?

Answer 2

1. Understand that if 𝑎^𝑦 = 𝑥 then 𝑦 = log_𝑎 (𝑥)
2. Use logarithms to solve for the exponent in an equation involving indices by taking the logarithm of both sides.

Question 3

What are the laws of logarithms, and how do you apply them?

Answer 3

1. Product: log_b (𝑥𝑦) = log_𝑏(𝑥) + log_⁡𝑏 (𝑦)
2. Quotient: log_𝑏 (𝑥/𝑦) = log_𝑏 (𝑥) − log_𝑏(𝑦)
3. Power: log_𝑏(𝑥ⁿ) = 𝑛 × log_𝑏 (𝑥)
4. Change of Base: log_𝑏 (𝑥) = log_𝑘(𝑥) / log_𝑘(𝑏) for any base 𝑘.

Question 4

How do you find the logarithm of any positive number to a given base?

Answer 4

1. Understand that log_𝑏(𝑥) for the exponent
   𝑦 such that 𝑏^𝑦 = 𝑥
2. Use a calculator or logarithm tables to find the value directly for common bases like 10 or 𝑒.
3. Apply change of base formula if the base is not common: log_𝑏(𝑥) = log_𝑐(𝑥) / log_𝑐 (𝑏) where 𝑐 is a base you can easily calculate.

Question 5

How do you change the base in logarithms, and how do you apply it?

Answer 5

Use the change of base formula:
log_𝑏 (𝑥) = log_𝑐(𝑥) / log_𝑐(𝑏), where 𝑐 is a convenient base, usually 10 or 𝑒.
Apply this formula to convert logarithms to a base that is easier to work with.

Question 6

How do you simplify and rationalize surds?

Answer 6

1. Simplify: Factorize the number under the root to find perfect squares and take them out of the root.
   Example: √50 = √(25 × 2) = 5 √2
2. Rationalize: Multiply the surd by a conjugate to remove the surd from the denominator.
   Example: 1/√2 × √2 / √2 = √2/2

Question 7

How do you perform basic operations (addition, subtraction, multiplication, division) on surds?

Answer 7

1. Addition/Subtraction: Combine like terms (similar surds).
   Example: 2√3 + 3√3 = 5√3
2. Multiplication: Multiply the coefficients and then the radicands.
   Example: √2 × √3 = √6
3. Division: Simplify by rationalizing the denominator if necessary.
   Example: √2 / √3 = √6 / 3