Radicals and Rational Exponents

Factoring by Grouping

• Multiply the first and last numbers, resulting in a product (e.g., negative 12).
• Identify factors of the product (e.g., factors of 12: 1, 2, 3, 4, 6, 12).
• Determine which factors, when added, yield the coefficient of the middle term (e.g., 3 and -4).

Principal Square Root

• The principal square root refers to the positive square root.
• When dealing with negative numbers inside the square root, we are looking for a number that, when multiplied by itself, results in that negative number (e.g., square root of negative 80).

Calculator Usage and Answer Verification

• Before inputting into a calculator, check if the question requires a fractional or decimal answer.
• Use the calculator to find square roots, specifically the x squared button (usually next to the number 1 on the left).
• The second function of the x squared button typically gives the square root symbol.
• After obtaining an answer, pause and verify if it makes sense in the given context.

Perfect Squares and Estimation

• Perfect squares are numbers that are the result of squaring an integer.
• To estimate the square root of a number, find the perfect squares it lies between.
• For example (\sqrt{20}) lies between (\sqrt{16} = 4) and (\sqrt{25} = 5), so the answer should be between 4 and 5.

(4^2 = 16)
(5^2 = 25)

Roots Beyond Square Roots

• The radical symbol without an index implies a square root.
• The index of the radical indicates the type of root (e.g., 3 for cube root, 4 for fourth root, 5 for fifth root).
• For a cube root, find a number that multiplies by itself three times to give the radicand. \sqrt[3]{x}
• Square roots of negative numbers do not yield real answers.
• Cube roots can have negative numbers; the result will share the sign (e.g., \sqrt[3]{-64} = -4).
• It's helpful to have a list of perfect squares and perfect cubes handy.

Examples of Cubes

• 1^3 = 1 * 1 * 1 = 1
• 2^3 = 2 * 2 * 2 = 8
• 3^3 = 3 * 3 * 3 = 27
• 4^3 = 4 * 4 * 4 = 64

Odd vs. Even Roots

• Odd roots (cube root, fifth root, etc.) can have negative numbers underneath the radical.
• Even roots (square root, fourth root, etc.) cannot have negative numbers underneath the radical.

Rational Exponents

• a^{\frac{1}{n}} is defined as the nth root of a.
• 64^{\frac{1}{3}} is the same as the cube root of 64, which equals 4.
• x^{\frac{1}{4}} is the same as the fourth root of x.

Rational Exponents with Additional Terms

• (81x^8)^{\frac{1}{4}} implies that both 81 and x^8 are raised to the power of \frac{1}{4}.
• In expressions like y^{\frac{1}{3}}, only y is being raised to that power if there are no parentheses.

Examples of Roots

• The cube root of 1000 is 10. \sqrt[3]{1000} = 10
• \sqrt[3]{x}
• The fourth root of 1 is 1. \sqrt[4]{1} = 1
• - \sqrt{64} = -8
• \sqrt[3]{125x^9} = 5x^3

Powers and Roots in Rational Exponents

• With rational exponents, the top number is the power, and the bottom number is the root.
• Remember: Flower (power) over root. The top number is the exponent (power), the bottom number is the root.

Example

• 4^{\frac{3}{2}} can be written as (\sqrt{4})^3 or (\sqrt{4^3}), both equal 8.
  • The order of applying the root and power doesn't matter.

Additional Example:

• For a^{\frac{3}{4}}, we can write it as (\sqrt[4]{a})^3 or \sqrt[4]{a^3}.
• Fourth root of 16 is 2, then 2^3 = 8.

Negative Numbers and Rational Exponents

• Negative numbers are inside the whole root and power expression.
• For (-27)^{\frac{2}{3}}, square the result once the cube is found.
• Cube root of -27 is -3, and (-3)^2 = 9.

Fractions and Rational Exponents

• (\frac{1}{9})^{\frac{3}{2}} becomes (\frac{\sqrt{1}}{\sqrt{9}})^3
• If there is more than one term on the inside, such as (4x - 1), then apply the exponent to the entire expression.