Advanced understanding of powers and roots extends beyond simple squares and cubes, encompassing a broader spectrum of mathematical operations. This section aims to deepen students' comprehension of these critical concepts, ensuring proficiency in handling a variety of numerical and algebraic expressions.

Introduction to Indices and Roots

Indices (or exponents) and roots represent fundamental mathematical operations. Indices indicate repeated multiplication of a base number, while roots seek the base number that, when raised to a specific power, yields the original number.

Indices: Beyond Basics

• Zero Exponent Rule: For any non-zero base $a$, $a^0 = 1$.
• Negative Exponents: Express $a^{-n}$ as $a^{-n} = \frac{1}{a^n}$, highlighting inverse relationships.
• Fractional Exponents: Understand $a^{\frac{m}{n}}$ as the nth root of $a^m$, or $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Expanding the Concept of Roots

Roots reverse the operation of exponentiation, with the square root and cube root being the initial steps into this inverse world.

• Square Roots: $\sqrt{x}$ finds a number that squared equals $x$.
• Cube Roots: $\sqrt[3]{x}$ identifies a number that cubed returns $x$.
• nth Roots: The nth root $\sqrt[n]{x}$ locates a number that, when raised to the nth power, equals $x$.

Advanced Laws of Indices

The manipulation and simplification of expressions with indices are governed by specific laws.

• Multiplication: Combine bases with the same exponent by adding their exponents: $a^m \times a^n = a^{m+n}$.
• Division: Divide bases by subtracting exponents: $a^m \div a^n = a^{m-n}$.
• Power of a Power: Multiply exponents when a power is raised to another power: $(a^m)^n = a^{mn}$.
• Power of a Product: Distribute the exponent over a product: $(ab)^n = a^n b^n$.
• Fractional Power: Apply the exponent to both numerator and denominator: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

Worked Examples

Example 1: Simplifying an Expression

Question: Simplify $(2^3)^2 \times 2^{-1}$.

Solution:

1. Power of a Power: $((2^3)^2 = 2^{3 \times 2} = 2^6)$
2. Applying Negative Exponent: $(2^6 \times 2^{-1} = 2^{6-1} = 2^5)$
3. Simplify: $(2^5 = 32)$

Example 2: Calculating Roots

Question: Calculate $\sqrt[3]{-27}$.

Solution:

1. Identify the Cube Root: The cube root of -27 is a number that, when cubed, equals -27.
2. Calculation: $(-3)^3 = -27$, hence, $\sqrt[3]{-27} = -3$.

Practice Questions

Question 1: Simplifying an Expression

Simplify the expression $\frac{4^3 \times 2^2}{8^2}$.

Solution:

• Calculate numerator and denominator separately:
  • Numerator: $4^3 \times 2^2 = 64 \times 4 = 256$
  • Denominator: $8^2 = 64$
• Simplify the fraction: $\frac{256}{64} = 4$

The simplified expression is $4.0$.

Question 2: Calculating with Fractional Exponents

Calculate $16^{\frac{1}{2}} \times 8^{\frac{1}{3}}$.

Solution:

1. Calculate each term:

• $16^{\frac{1}{2}} = \sqrt{16} = 4$
• $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$

2. Multiply the results: $4 \times 2 = 8$

The calculated result is $8$.

Question 3: Solving for an Exponent

Find the value of $x$ in the equation $5^{2x} = 625$.

Solution:

1. Recognise that $625 = 5^4$.

2. Set up the equation: $5^{2x} = 5^4$.

3. Since the bases are the same, the exponents must be equal: $2x = 4$.

4. Solve for $x$: $x = 2$.

The value of $x$ that satisfies the equation is $2$. Note: The solution involving $\log(25) + I\pi$ is outside the real number scope and not relevant for this context.