Indices and Standard Form - Form 3

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Indices and Standard Form - Form 3

Laws of indices, scientific notation, and applications.

📘 Topic Summary

Indices and Standard Form are fundamental concepts in mathematics that help us simplify complex expressions and perform calculations more efficiently. This study guide will cover the laws of indices, scientific notation, and their applications.

📖 Glossary

Index: A small number that shows how many times a base number should be multiplied by itself.
Base: The number being multiplied by itself in an expression with indices.
Scientific Notation: A way of writing very large or very small numbers using exponents and coefficients.
Exponent: The power to which a base is raised in an expression with indices.

⭐ Key Points

Indices can be used to simplify expressions by combining like terms.
Scientific notation helps us write very large or very small numbers in a more manageable way.
Understanding the laws of indices and scientific notation is crucial for solving problems in mathematics and science.
The power of an expression with indices can be changed by adding or subtracting the same value to both the base and exponent.
When simplifying expressions, it's essential to follow the order of operations (PEMDAS).

🔍 Subtopics

Introduction to Indices

Indices are a shorthand way of writing repeated multiplication. For example, the expression 2^3 can be read as '2 to the power of 3' and is equivalent to 2 multiplied by itself three times: 2 × 2 × 2 = 8. The index or exponent tells us how many times to multiply the base number.

Simplifying Expressions with Indices

When simplifying expressions with indices, we can use the commutative and associative properties of multiplication. For example, 2^3 × 2^5 = 2^(3+5) = 2^8. We can also combine like terms by adding or subtracting the indices: 2^3 + 2^3 = 2^(3+3) = 2^4.

Scientific Notation

Scientific notation is a way of writing very large or very small numbers in a more manageable form. It involves multiplying the number by 10 raised to a power, for example: 4500 can be written as 4.5 × 10^3. This makes it easier to compare and perform calculations with these types of numbers.

Real-World Applications of Indices and Standard Form

Indices are used in many real-world applications, such as calculating the area and volume of shapes, modeling population growth, and determining the effects of compound interest. Scientific notation is useful for expressing very large or small measurements, like distances to stars or sizes of atoms.

Common Mistakes to Avoid

When working with indices, it's easy to make mistakes by forgetting to simplify expressions or incorrectly applying the commutative and associative properties. Make sure to follow the order of operations (PEMDAS) and double-check your calculations.

Practice Problems

Simplify each expression: 3^2 × 3^4, 2^5 + 2^3, 10^2 - 5^2. Express the following numbers in scientific notation: 3400, 0.000045.

Conclusion

In this section, we have explored the basics of indices and scientific notation. By mastering these concepts, you will be able to simplify expressions and work with very large or small numbers in a more efficient way.

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