Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Number Systems ↓


Exponents and Radicals


In mathematics, it is important to understand how numbers interact and are manipulated. In these interactions, exponents and radicals emerge as central concepts in algebra and beyond. They allow us to express repeated multiplication and roots in a concise way. Let us understand these concepts one by one.

Exponential

Exponentiation refers to the number, known as the base, that is multiplied by itself. It is a powerful way to express repeated multiplication in a concise form. For example, when you see an expression like 2^5, it means that the number 2 is to be multiplied by itself 5 times, as shown below:

    2^5 = 2 × 2 × 2 × 2 × 2

In this expression, 2 is the base, and 5 is the exponent.

The general form of exponents can be written as: a^n where a is the base, and n is the exponent.

Basic properties of exponents

  • Product of powers: When multiplying like bases, we add the exponents. a^m × a^n = a^(m+n)
  • Example: 3^2 × 3^3 = 3^(2+3) = 3^5
  • Power of a power: When raising a power to another power, multiply the exponents. (a^m)^n = a^(m*n)
  • Example: (2^3)^2 = 2^(3*2) = 2^6
  • Power of a product: Distribute the exponent over all the factors inside the parentheses. (ab)^n = a^n × b^n
  • Example: (3 × 4)^2 = 3^2 × 4^2 = 9 × 16
  • Zero exponent: any non-zero number raised to the power of zero equals 1 a^0 = 1 (if a ≠ 0)
  • Example: 5^0 = 1
  • Negative Exponents: A negative exponent represents the inverse of the base raised to the absolute value of the exponent. a^(-n) = 1 / a^n
  • Example: 2^-3 = 1/(2^3) = 1/8
Example view:2^3 = 2 × 2 × 2 = 8a^0 = 1 (a ≠ 0)

Radicals

Radical sign refers to (sqrt{}{}), which is used to represent the root of numbers. The most common radical is the square root. Finding the square root is the opposite of squaring a number. If the square of a number is a given value, then the square root of that value is the radical number.

Square root

The square root of a number is the value that when multiplied by itself gives the original number. It can be represented as √a, which tells us that a number when multiplied by itself gives a.

    Example: √9 = 3 because 3 × 3 = 9
Square root example visualization:√16 = 4 because 4 × 4 = 16

Cube root

When working with cube roots, the idea is the same, but instead of multiplying the same number twice, you multiply it three times. This is represented as ∛a, asking what number multiplied three times gives a.

    Example: ∛27 = 3 because 3 × 3 × 3 = 27

Properties of radicals

  • Product Property: The square root of a product is equal to the product of the square roots of each factor. √(ab) = √a × √b
  • Example: √(16 × 25) = √16 × √25 = 4 × 5 = 20
  • Quotient Property: The square root of the quotient is equal to the quotient of the square root of the numerator and denominator. √(a/b) = √a / √b
  • Example: √(4/9) = √4 / √9 = 2/3
  • Radicals with exponents: You can also express roots as exponents. The nth root of a number can be written with a fractional exponent. a^(1/n) = √[n]{a}
  • Example: 27^(1/3) = ∛27 = 3
Cube root example visualization:∛64 = 4 because 4 × 4 × 4 = 64

Combining Exponents and Radicals

Sometimes, in mathematical problems, you will need to use both exponents and radicals. Here we will explore how they interact and how they can be used together.

Simplifying Expressions with Exponents and Radicals

An expression such as (x^2)^(1/2) can be simplified using the properties mentioned previously.

    (x^2)^(1/2) = x^(2 * (1/2)) = x^1 = x

This shows that extracting a square root and squaring are opposite operations that "cancel out" each other.

Rationalizing the denominator

When a fraction has a radical in the denominator, it is often better to remove the radical from the denominator. This process is called "rationalizing the denominator."

  • Basic Rationalisation: 
            1/√2 = 1/√2 × √2/√2 = √2/2
  • Example with binomials: 
            1/(√2 + √3) = 1/(√2 + √3) × (√2 - √3)/(√2 - √3) = (√2 - √3)/(2 - 3)
        = - (√2 - √3)

Conclusion

In the study of number systems, exponents and radicals serve as essential tools that simplify and expand our mathematical expressions. They are indispensable in algebra and provide a stepping stone to more advanced mathematical concepts such as logarithms and calculus.

Through understanding the properties and applications of exponents and radicals, students gain the ability to solve and manipulate complex equations, further enhancing their mathematical proficiency and preparing them for future mathematical challenges.


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Exponents and Radicals

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