Problem 10 - Algebra 1: Concepts and Skills

Question

Rewrite the expression using rational exponent notation. $$ \sqrt{14} $$

Step-by-Step Solution

Answer

\( 14^{\frac{1}{2}} \)

Step 1: Conversion to Rational Exponent

The square root of any number, say n, can be expressed in the rational exponent notation as \( n^{\frac{1}{2}} \). Using this rule, the square root of 14 in rational exponent notation can be written as \( 14^{\frac{1}{2}} \).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \). For example, if you're asked to find the square root of 9, you would determine that 3 multiplied by itself is 9, so \( \sqrt{9} = 3 \). When it comes to irrational numbers, which cannot be expressed as a simple fraction, square roots are often left in radical form. For instance, \( \sqrt{14} \) cannot be simplified into a neat fraction, as 14 is not a perfect square. Thus, it remains \( \sqrt{14} \) as an exact value, unless we approximate to a decimal or use rational exponent notation to rewrite it.

Exponents

Exponents are used in algebra to denote how many times a number, known as the base, is multiplied by itself. The exponent is written as a small number to the top right of the base. For example, \(4^3\) means that 4 is multiplied by itself 3 times: \(4 \times 4 \times 4\), which equals 64. Rational exponents are a way of representing roots and are particularly helpful with irrational numbers. They are expressed as fractions, where the numerator (top part) is the power and the denominator (bottom part) is the root. So, when you see \(14^{\frac{1}{2}}\), it is equivalent to the square root of 14. This notation is very useful for further algebraic operations, as it smoothly integrates into the laws of exponents that govern multiplication, division, and raising to powers of powers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical ideas. They do not contain an equal sign, which distinguishes them from equations. Variables in these expressions stand for unknown values that can be manipulated through algebraic principles. For instance, the expression \(2x + 3\) has a variable \(x\), a coefficient (2), and a constant (3).

In the context of exponents and roots, algebraic expressions can include variables within radical signs or with rational exponents. An example could be \(x^{\frac{1}{3}}\), which represents the cube root of \(x\). Understanding how to handle these expressions empowers students to tackle complex problems by applying a mixture of arithmetic and algebraic rules. It's the first step towards solving equations and understanding the relationships between various algebraic terms.