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OCR GCSE Maths (Higher) Study Notes

1.3.2 Advanced Powers and Roots

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 aa, a0=1a^0 = 1.
  • Negative Exponents: Express ana^{-n} as an=1ana^{-n} = \frac{1}{a^n}, highlighting inverse relationships.
  • Fractional Exponents: Understand amna^{\frac{m}{n}} as the nth root of ama^m, or amn=amna^{\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: x\sqrt{x} finds a number that squared equals xx.
  • Cube Roots: x3\sqrt[3]{x} identifies a number that cubed returns xx.
  • nth Roots: The nth root xn\sqrt[n]{x} locates a number that, when raised to the nth power, equals xx.

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: am×an=am+na^m \times a^n = a^{m+n}.
  • Division: Divide bases by subtracting exponents: am÷an=amna^m \div a^n = a^{m-n}.
  • Power of a Power: Multiply exponents when a power is raised to another power: (am)n=amn(a^m)^n = a^{mn}.
  • Power of a Product: Distribute the exponent over a product: (ab)n=anbn(ab)^n = a^n b^n.
  • Fractional Power: Apply the exponent to both numerator and denominator: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.

Worked Examples

Example 1: Simplifying an Expression

Question: Simplify (23)2×21(2^3)^2 \times 2^{-1}.

Solution:

  1. Power of a Power: ((23)2=23×2=26)((2^3)^2 = 2^{3 \times 2} = 2^6)
  2. Applying Negative Exponent: (26×21=261=25)(2^6 \times 2^{-1} = 2^{6-1} = 2^5)
  3. Simplify: (25=32)(2^5 = 32)

Example 2: Calculating Roots

Question: Calculate 273\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(-3)^3 = -27, hence, 273=3\sqrt[3]{-27} = -3.

Practice Questions

Question 1: Simplifying an Expression

Simplify the expression 43×2282\frac{4^3 \times 2^2}{8^2}.

Solution:

  • Calculate numerator and denominator separately:
    • Numerator: 43×22=64×4=2564^3 \times 2^2 = 64 \times 4 = 256
    • Denominator: 82=648^2 = 64
  • Simplify the fraction: 25664=4\frac{256}{64} = 4

The simplified expression is 4.04.0.

Question 2: Calculating with Fractional Exponents

Calculate 1612×81316^{\frac{1}{2}} \times 8^{\frac{1}{3}}.

Solution:

1. Calculate each term:

  • 1612=16=416^{\frac{1}{2}} = \sqrt{16} = 4
  • 813=83=28^{\frac{1}{3}} = \sqrt[3]{8} = 2

2. Multiply the results: 4×2=84 \times 2 = 8

The calculated result is 88.

Question 3: Solving for an Exponent

Find the value of xx in the equation 52x=6255^{2x} = 625.

Solution:

1. Recognise that 625=54625 = 5^4.

2. Set up the equation: 52x=545^{2x} = 5^4.

3. Since the bases are the same, the exponents must be equal: 2x=42x = 4.

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

The value of xx that satisfies the equation is 22. Note: The solution involving log(25)+Iπ\log(25) + I\pi is outside the real number scope and not relevant for this context.

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