Powers & Exponents, Roots & Radicals, with Polynomials
This lesson examines exponents and how they relate to square roots and cube roots. It also discusses powers of 10 and polynomials.
Powers
We have explained in previous lessons that exponents imply an expression of repeated multiplication, otherwise known as a power. There are many rules associated with exponents. Recall that any number to a power of 0 will equal 1, or \(n^0 = 1\). Additionally, the negative exponent rule reveals that numbers with a negative power are equal to a fraction.
Keep In Mind
Most square roots and cube roots are not perfect roots.
Powers can express repeated multiplication, and radicals operate to undo that multiplication. 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 \(\sqrt{100}=10\). 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 \(pm \sqrt{p}\) because a negative value squared is a positive solution.
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 100, 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 \(\sqrt[3]{p}=x\).
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 \(x^2 = 2\).
quares and cubes.
| \(1^2 = 1\) | \(\sqrt{1}=1\) | \(7^2 =49\) | \(\sqrt{49}=7\) |
| \(2^2 = 4\) | \(\sqrt{4}=2\) | \(8^2 = 64\) | \(\sqrt{64}=8\) |
| \(3^2 = 9\) | \(\sqrt{9}=3\) | \(9^2 = 81\) | \(\sqrt{81}=9\) |
| \(4^2 = 16\) | \(\sqrt{16}=4\) | \(10^2 = 100\) | \(\sqrt{100}=10\) |
| \(5^2 = 25\) | \(\sqrt{25}=5\) | \(11^2 = 121\) | \(\sqrt{121}=11\) |
| \(6^2 = 36\) | \(\sqrt{36}=6\) | \(12^2 = 144\) | \(\sqrt{144}=12\) |
| \(1^3= 1\) | \(\sqrt[3]{1}=1\) | \(7^3=343\) | \(\sqrt[3]{343}=7\) |
| \(2^3=8\) | \(\sqrt[3]{8}=2\) | \(8^3=512\) | \(\sqrt[3]{512}=8\) |
| \(3^3=27\) | \(\sqrt[3]{27}=3\) | \(9^3=729\) | \(\sqrt[3]{729}=9\) |
| \(4^3=64\) | \(\sqrt[3]{64}=4\) | \(10^3=1000\) | \(\sqrt[3]{1000}=10\) |
| \(5^3=125\) | \(\sqrt[3]{125}=5\) | ||
| \(6^3=216\) | \(\sqrt[3]{216}=6\) |
Powers Review
Understanding Exponents
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 \(3 \times 10^4\) is the same as 3 × 10,000 = 30,000.
The number \(3 \times 10^{-4}\) is the same as 3 × 0.0001 = 0.0003.
For example, the population of the United States is about \(3 \times10^8\), and the population of the world is about \(7 \times 10^9\). 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.
Understanding Exponents Review
Polynomials
A polynomial is an expression that contains exponents, as well as variables, constants, and operations. The exponents of the variables are only whole numbers, and there is no division by a variable. The operations are addition, subtraction, multiplication, and division. Constants are terms without a variable. A polynomial of one term is a monomial; a polynomial of two terms is a binomial; and a polynomial of three terms is a trinomial.
To add polynomials, combine like terms and write the solution from the term with the highest exponent to the term with the lowest exponent. To simplify, first rearrange and group like terms. Next, combine like terms.
\((3x^2 + 5x – 6) + (4x^3  – 3x + 4)\)
\( = 4x^3 + 3x^2 + (5x – 3x) + (–6 + 4)\)
\( = 4x^3 + 3x^2 + 2x – 2\)
Keep In Mind
The solution is an expression, and a value is not calculated for the variable.
To subtract polynomials, rewrite the second polynomial using an additive inverse. Change the minus sign to a plus sign, and change the sign of every term inside the parentheses. Then, add the polynomials.
\((3x^2 + 5x – 6) – (4x^3  – 3x + 4) \)
\( = (3x^2 + 5x – 6) + (–4x^3 + 3x – 4)\)
\(= –4x^3 + 3x^2 + (5x + 3x) + (–6 – 4)\)
\( = –4x^3 + 3x^2 + 8x – 10\)
Using Polynomial Identities
There are many polynomial identities that show relationships between expressions Addition and subtraction expressions can be squared, and the square of a binomial expression can be written in two different forms:
- \((a + b)2 = a^2 + 2ab + b^2\)
- \((a – b)^2 = a^2  – 2ab + b^2\)
A simple way for solving a binomial is to remember to FOIL the expression. The square implies to repeat the multiplication of the operation. To solve the expression would require multiplying the First, Outside, Inside, and then Last variables which results in the relationship of the expressions.
Be Careful
Pay attention to the details of each variable and operation.
Familiarizing yourself with perfect squares is useful for dealing with exponents, and familiarizing yourself with the square of a binomial can similarly help in identifying binomials.
Polynomials Review
Let’s Review!
- Powers represent repeated multiplication and radicals are the inverse operation of exponents.
- Numbers expressed in scientific notation are useful to compare large or small numbers.
- Polynomials are expressions that include exponents, as well as variables, constants, and operations.
- Polynomials can be added, subtracted, and even squared.
- Squared polynomials are binomial expressions that can be written in different forms.
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