key term - Radicals

Definition

Radicals, in the context of simplifying expressions, refer to the square root symbol (√) and the nth root symbol (∛) used to represent the process of extracting roots of a number or expression. These radical symbols indicate the inverse operation of raising a number to a power.

5 Must Know Facts For Your Next Test

1. Radicals can be simplified by applying the properties of exponents, such as $\sqrt{a^n} = a^{n/2}$ and $\sqrt[n]{a^m} = a^{m/n}$.
2. Radicals can be combined using the addition or subtraction of like radicals, where the radicands are the same.
3. Radicals can be multiplied or divided by applying the product or quotient rule, which involves multiplying or dividing the radicands and adding or subtracting the exponents.
4. Rational exponents can be used to represent radicals, where $a^{1/n} = \sqrt[n]{a}$.
5. Radicals can be rationalized by multiplying the numerator and denominator by a conjugate expression to eliminate the radical in the denominator.

Review Questions

• Explain the process of simplifying an expression with a square root, such as $\sqrt{75}$.
  • To simplify an expression with a square root, such as $\sqrt{75}$, we first need to identify the largest perfect square that divides the radicand (75 in this case). In this example, the largest perfect square is 25. We can then rewrite the expression as $\sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}$. This simplifies the radical expression by separating the perfect square and the remaining factor under the radical.
• Describe the steps to rationalize the denominator of a fraction with a radical, such as $\frac{1}{\sqrt{2}}$.
  • To rationalize the denominator of a fraction with a radical, such as $\frac{1}{\sqrt{2}}$, we need to multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of $\sqrt{2}$ is $\sqrt{2}$. Multiplying the fraction by $\frac{\sqrt{2}}{\sqrt{2}}$ gives us $\frac{\sqrt{2}}{2}$. This eliminates the radical in the denominator and simplifies the expression.
• Analyze the relationship between rational exponents and radicals, and explain how to convert between them, such as expressing $\sqrt[3]{27}$ using a rational exponent.
  • Radicals and rational exponents are closely related, as they represent the same mathematical concept. The relationship can be expressed as $a^{1/n} = \sqrt[n]{a}$. To convert a radical expression to a rational exponent, we can use this relationship. For example, to express $\sqrt[3]{27}$ using a rational exponent, we can write it as $27^{1/3}$. This demonstrates the equivalence between the cube root of 27 and the expression with a rational exponent of 1/3, which both represent the same value.

Related terms

Square Root:

The square root of a number is the value that, when multiplied by itself, gives the original number. It is denoted by the symbol √.

Nth Root: The nth root of a number is the value that, when raised to the power of n, gives the original number. It is denoted by the symbol ∛ for the cube root, ∜ for the fourth root, and so on.

Radicand:

The radicand is the number or expression under the radical symbol, which is the value being operated on by the root.