Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Exponents and Powers ↓


Standard Form


In mathematics, the representation and handling of numbers is a fundamental concept. As numbers become very large or very small, working with them directly can become cumbersome and prone to errors. To ease this problem, mathematicians have devised systems such as exponents and powers, and one of the representations in this system is known as the “standard form.” In this detailed exploration, we will dive into the concept of standard form, specifically for the grade 8 level, and try to make it as simple as possible to understand.

What is a standard form?

Standard form is a way to write numbers that are too large or too small to write in decimal form. This form is especially useful in science and engineering to simplify the interpretation of numbers by representing them as powers of ten. The general format of a number in standard form is:

a × 10 n

Here:

  • a is a number that is greater than or equal to 1 and less than 10.
  • n is an integer, which means it can be a positive or negative whole number.

Example 1: Large numbers

To write the number 5,000,000 in standard form, follow these steps:

  1. Identify the number of digits before the decimal point.
  2. Move the decimal point so that there is only one non-zero digit to its left.
  3. Count the number of times the decimal point moves. This will be your exponent n.
5,000,000 = 5.0 × 10 6

Explanation: The decimal point in 5,000,000 is originally at the end. Moving it 6 places to the left puts it after the 5, making the exponent 6.

5,000,000 5.0 x 10 6

Example 2: Small numbers

Now consider a small number, such as 0.00034. To express this number in standard form:

  1. Move the decimal point to the right until there is only one non-zero digit to its left.
  2. Count the number of places shifted as a negative exponent.
0.00034 = 3.4 × 10 -4

Explanation: The decimal point moves 4 places to the right and hence the exponent is -4.

0.00034 3.4 x 10 -4

Conversion to standard form: Step-by-step

Step 1: Identify significant figures

This is the part of the number that has the actual value. For example, the significant figures in 4,570,000 are 4, 5, 7.

Step 2: Place the decimal point correctly

Place the decimal point so that there is only one digit to the left.

4570000 → 4.57

Step 3: Count the shifts

Determine how many places the decimal point was moved. This calculation becomes the exponent.

Step 4: Apply powers of ten

Write the number as a product of power tens: 4.57 × 10 6

Understanding powers of ten

Before we dive deeper into standard form examples, it's important to have a good understanding of powers of ten. The number ten is the base in our decimal number system, and it appears widely in standard form.

Powers of ten basics

When ten is raised to a positive exponent, you move the decimal point to the right, making the number larger. Conversely, a negative exponent moves the decimal point to the left, making the number smaller.

10 3 = 1,000 (three places to the right)
10 -3 = 0.001 (three places from the left)

More examples

Example 3

Suppose you want to write the number 9,870,000 in standard form. The significant figures are 9.87. Moving the decimal point 6 places gives:

9,870,000 = 9.87 × 10 6

Example 4

For a small number like 0.0000072:

0.0000072 = 7.2 × 10 -6

Here, the decimal moves 6 places to the right producing an exponent of -6.

Example 5

A very large number 780,000,000,000 would be written like this:

780,000,000,000 = 7.8 × 10 11

Example 6

For a small number 0.000000054, this becomes:

0.000000054 = 5.4 × 10 -8

Why use the standard form?

Using the standard form offers several advantages, especially when working with scientific calculations and large datasets:

  • Simplification: Standard Form simplifies arithmetic operations such as multiplication, division, addition, and subtraction when dealing with very large or very small numbers.
  • Efficiency: In science and engineering, it improves the efficiency of calculating values and makes calculations easier to manage, understand, and compare.
  • Accuracy: Maintains accuracy and precision by avoiding rounding errors in significant digits.

Special cases and considerations

When using the standard form, consider these special cases:

  • When n = 0, the value is exactly the number a. This happens with any number that can be represented as a single digit, or when the decimal and significant figures coincide at 1 × 10 0.
  • A negative n value represents numbers less than 1, while a positive n value represents numbers greater than 1.
  • Exact numbers, such as 1 in 1 × 10 n, must be represented without shifting the decimal point (i.e., no unnecessary decimal places).

Practice problems

Solving practice problems will help you solidify your understanding of standard form. Try expressing these numbers in standard form:

  1. 350,000
  2. 0.00000345
  3. 6,570
  4. 0.452
  5. 9,999,999,999
  6. 0.098765

Solution:

  1. 350,000 = 3.5 × 10 5
  2. 0.00000345 = 3.45 × 10 -6
  3. 6,570 = 6.57 × 10 3
  4. 0.452 = 4.52 × 10 -1
  5. 9,999,999,999 = 9.999999999 × 10 9
  6. 0.098765 = 9.8765 × 10 -2

Standard form is an essential concept, not only in mathematics but also in various scientific disciplines. The ability to efficiently manipulate numbers and properly understand their magnitude is one of the key skills developed through such mathematical exploration.


Grade 8 → 9.3


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Negative Exponents
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Standard Form

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