Lesson 1.4: Powers, Roots, and Standard Form
Introduction
In this lesson, we will explore the concepts of powers, roots, and standard form, which lay the groundwork for many mathematical operations you'll encounter later. Understanding these concepts is critical for performing calculations accurately and efficiently, both with and without a calculator. By the end of this lesson, you will be able to:
- Describe squares, cubes, higher powers, and their roots.
- Explain the laws of indices for positive integer powers and the meaning of a zero index.
- Use standard form (scientific notation) for very large and very small numbers.
- Evaluate powers and roots of numbers.
- Apply the basic laws of indices to simplify numerical expressions.
1. Squares, Cubes, and Higher Powers
1.1. Squares
The square of a number is the result of multiplying that number by itself. If we denote a number as $x$, then its square is expressed as:
$$x^2 = x \times x$$
Example 1:
Calculate the square of 5.
$$5^2 = 5 \times 5 = 25$$
So, the square of 5 is 25.
1.2. Cubes
Similarly, the cube of a number is the result of multiplying that number by itself twice. It can be represented as:
$$x^3 = x \times x \times x$$
Example 2:
Find the cube of 3.
$$3^3 = 3 \times 3 \times 3 = 27$$
Thus, the cube of 3 is 27.
1.3. Higher Powers
Higher powers are simply the result of multiplying a number by itself more than three times. For instance, the fourth power of $x$ is given by:
$$x^4 = x \times x \times x \times x$$
Example 3:
Calculate the fourth power of 2.
$$2^4 = 2 \times 2 \times 2 \times 2 = 16$$
Hence, the fourth power of 2 is 16.
2. Roots of Numbers
A root is essentially the inverse operation of raising a number to a power. The square root of $x$ is the number that, when multiplied by itself, results in $x$. It is denoted as:
$$\sqrt{x}$$
Similarly, the cube root is the number that, when used three times in multiplication, gives $x$ and is denoted as:
$$\sqrt[3]{x}$$
Example 4:
Find the square root of 36.
$$\sqrt{36} = 6$$
Because $6 \times 6 = 36$.
Example 5:
Determine the cube root of 27.
$$\sqrt[3]{27} = 3$$
Since $3 \times 3 \times 3 = 27$.
3. The Laws of Indices
The laws of indices, also known as the laws of exponents, provide a set of rules for manipulating expressions involving powers. Here are the most important rules:
3.1. Multiplication of Powers
When you multiply powers with the same base, you add the exponents:
$$a^m \times a^n = a^{m+n}$$
Example 6:
Solve $2^3 \times 2^2$.
$$2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$$
3.2. Division of Powers
When dividing powers with the same base, you subtract the exponents:
$$\frac{a^m}{a^n} = a^{m-n}$$
Example 7:
Calculate $5^4 \div 5^2$.
$$\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25$$
3.3. Power of a Power
When you raise a power to another power, you multiply the exponents:
$$(a^m)^n = a^{m \cdot n}$$
Example 8:
Evaluate $(3^2)^3$.
$$(3^2)^3 = 3^{2 \cdot 3} = 3^6 = 729$$
3.4. Zero Index
Any non-zero number raised to the power of zero is equal to 1:
$$a^0 = 1 \quad (a
eq 0)$$
Example 9:
What is $7^0$?
$$7^0 = 1$$
4. Standard Form
Standard form, or scientific notation, is a way of expressing very large or very small numbers conveniently. A number is written in standard form as:
$$a \times 10^n$$
where $1 \leq a < 10$, and $n$ is an integer. This form helps simplify arithmetic operations involving large numbers.
4.1. Example of a Large Number
Convert 3,000,000 into standard form.
$$3,000,000 = 3 \times 10^6$$
4.2. Example of a Small Number
Convert 0.00045 into standard form.
$$0.00045 = 4.5 \times 10^{-4}$$
Conclusion
In this lesson, we covered the fundamental aspects of powers, roots, and standard form, along with the laws of indices. Mastery of these concepts is vital to understanding higher mathematics and performing calculations efficiently. Be sure to practice these skills to gain confidence in your computational abilities.
Study Notes
- A square is a number multiplied by itself ($x^2 = x \times x$).
- A cube is a number multiplied by itself three times ($x^3 = x \times x \times x$).
- Roots are the reverse operation of exponents (e.g., $\sqrt{36} = 6$).
- The laws of indices simplify calculations with powers:
- Multiplying powers: $a^m \times a^n = a^{m+n}$.
- Dividing powers: $\frac{a^m}{a^n} = a^{m-n}$.
- Power of a power: $(a^m)^n = a^{m \cdot n}$.
- Zero index: $a^0 = 1$.
- Standard form is used for large and small numbers, expressed as $a \times 10^n$.