Powers, Exponents, Roots, and Radicals
This lesson introduces how to apply the properties of exponents and examines square roots and cube roots. It also discusses how to estimate quantities using integer powers of 10.
Properties of Exponents
An expression that is a repeated multiplication of the same factor is a power. The exponent is the number of times the base is multiplied. For example, \(6^2\) is the same as 6 times 6, or 36. There are many rules associated with exponents.
| Property | Definition | Examples |
| Product Rule (Same Base) | \(a^m × a^n = a^{m+n}\) | \(4^1 × 4^4 = 4^{1+4} = 4^5 = 1024\\x^1 × x^4 = x^{1+4} = x^5\) |
| Product Rule (Different Base) | \(a^m × b^m = (a × b)^m\) | \(2^2 × 3^2 = (2 × 3)^2 = 6^2 = 36\\3^3 × x^3 = (3 × x)^3 = (3x)^3 = 27 x^3\) |
| Quotient Rule (Same Base) | \(\frac{a^m}{a^n} = a^{m–n}\) | \(\frac{4^4}{4^2} = 4^{4–2} = 4^2 = 16\\ \frac{x^6}{x^3} = x^{6–3} = x^3\) |
| Quotient Rule (Different Base) | \(\frac{a^m}{b^m} = (\frac{a}{b})^m\) | \(\frac{4^4}{3^4} = (\frac{4}{3})^4\\ \frac{x^6}{y^6} = (\frac{x}{y})^6\) |
| Power of a Power Rule | \((a^m)^n = a^{mn}\) | \((2^2)^3 = 2^{2×3} = 2^6 = 64\\ (x^5)^8 = x^{5×8} = x^{40}\) |
| Zero Exponent Rule | \(a^0 = 1\) | \(64^0 = 1\\ y^0 = 1\) |
| Negative Exponent Rule | \(a^{–m} = \frac{1}{a^m}\) | \(3^{–3} = \frac{1}{3^3} = \frac{1}{27}\\ \frac{1}{x^{–3}} = x^3\) |
Keep In Mind
The expressions \((–2)^2 = (–2) × (–2) = 4\) and \(–2^2 = –(2 × 2) = –4\) have different results because of the location of the negative signs and parentheses. For each problem, focus on each detail to simplify completely and correctly.
For many exponent expressions, it is necessary to use multiplication rules to simplify the expression completely.
Examples
Properties of Exponents Review
Square Roots and Cube Roots
The square of a number is the number raised to the power of 2. The square root of a number, when the number is squared, gives that number. \(10^2 = 100\), so the square of 100 is 10, or 100 = 10. Perfect squares are numbers with whole number square roots, such as 1, 4, 9, 16, and 25.
Squaring a number and taking a square root are opposite operations, meaning that the operations undo each other. This means that \(\sqrt{x^2} = x\) and \((\sqrt{x})^2 = x\). When solving the equation \(x^2 = p\), the solutions are \(x = ±\sqrt{p}\) because a negative value squared is a positive solution.
Keep In Mind
Most square roots and cube roots are not perfect roots.
The cube of a number is the number raised to the power of 3. The cube root of a number, when the number is cubed, gives that number. \(10^3 = 1000\), so the cube of 1,000 is 10, or \(\sqrt[3]{1000} = 10\). Perfect cubes are numbers with whole number cube roots, such as 1, 8, 27, 64, and 125.
Cubing a number and taking a cube root are opposite operations, meaning that the operations undo each other. This means that \(\sqrt[3]{x^3} = x \) and \((\sqrt[3]{x})^3 = x\). When solving the equation \(x^3 = p\), the solution is \(x = \sqrt[3]{p}\).
If a number is not a perfect square root or cube root, the solution is an approximation. When this occurs, the solution is an irrational number. For example, \(\sqrt{2}\) is the irrational solution to \(x^2 = 2\).
Below is a table of commonly known perfect squares and cubes.
| \(1^2 = 1\) | \(\sqrt{1}=1\) | \(7^2 =49\) | \(\sqrt{49}=7\) |
| \(2^2 = 4\) | \(\sqrt{4}=2\) | \(8^2 = 64\) | \(\sqrt{64}=8\) |
| \(3^2 = 9\) | \(\sqrt{9}=3\) | \(9^2 = 81\) | \(\sqrt{81}=9\) |
| \(4^2 = 16\) | \(\sqrt{16}=4\) | \(10^2 = 100\) | \(\sqrt{100}=10\) |
| \(5^2 = 25\) | \(\sqrt{25}=5\) | \(11^2 = 121\) | \(\sqrt{121}=11\) |
| \(6^2 = 36\) | \(\sqrt{36}=6\) | \(12^2 = 144\) | \(\sqrt{144}=12\) |
| \(1^3= 1\) | \(\sqrt[3]{1}=1\) | \(7^3=343\) | \(\sqrt[3]{343}=7\) |
| \(2^3=8\) | \(\sqrt[3]{8}=2\) | \(8^3=512\) | \(\sqrt[3]{512}=8\) |
| \(3^3=27\) | \(\sqrt[3]{27}=3\) | \(9^3=729\) | \(\sqrt[3]{729}=9\) |
| \(4^3=64\) | \(\sqrt[3]{64}=4\) | \(10^3=1000\) | \(\sqrt[3]{1000}=10\) |
| \(5^3=125\) | \(\sqrt[3]{125}=5\) | ||
| \(6^3=216\) | \(\sqrt[3]{216}=6\) |
Examples
Square Roots and Cube Roots Review
Express Large or Small Quantities as Multiples of 10
Scientific notation is a large or small number written in two parts. The first part is a number between 1 and 10. In these problems, the first digit will be a single digit. The number is followed by a multiple to a power of 10. A positive integer exponent means the number is greater than 1, while a negative integer exponent means the number is smaller than 1. Negative exponents are commonly used to represent small decimal numbers, but positive exponents can also be used to represent larger values.
Keep In Mind
A positive exponent in scientific notation represents a large number, while a negative exponent represents a small number.
The number \(3 × 10^4\) is the same as 3 × 10,000 = 30,000.
The number \(3 × 10^{–4}\) is the same as 3 × 0.0001 = 0.0003.
For example, the population of the United States is about \(3 × 10^8\), and the population of the world is about \(7 × 10^9\). The population of the United States is 300,000,000, and the population of the world is 7,000,000,000. The world population is about 20 times larger than the population of the United States.
Examples
Express Large or Small Quantities as Multiples of 10 Review
Let’s Review!
- The properties and rules of exponents are applicable to generate equivalent expressions.
- Only a few whole numbers out of the set of whole numbers are perfect squares. Perfect cubes can be positive or negative.
- Numbers expressed in scientific notation are useful to compare large or small numbers.
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