Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-52: Rewrite the expression using rational exponents. $$ \sqrt{a+\sqrt{b}} $$

Step-by-Step Solution

Answer

The expression \( \sqrt{a + \sqrt{b}} \) can be rewritten as \( (a + b^{1/2})^{1/2} \).

Step 1: Understand the Expression Inside the Square Root

First, identify the different components in the expression \( \sqrt{a + \sqrt{b}} \). Notice that inside the square root, there is an expression \( a + \sqrt{b} \).

Step 2: Rewrite Inner Square Root

Rewrite the inner square root \( \sqrt{b} \) using rational exponent notation. The square root can be written as an exponent of \( \frac{1}{2} \). Hence, \( \sqrt{b} = b^{1/2} \).

Step 3: Substitute Rational Exponent into the Expression

Substitute the expression for \( \sqrt{b} \) back into the original expression. This gives us \( \sqrt{a + b^{1/2}} \).

Step 4: Rewrite the Entire Expression Using Rational Exponents

Now, rewrite the entire expression, \( \sqrt{a + b^{1/2}} \), using a rational exponent. The outer square root can also be written with an exponent of \( 1/2 \). Thus, the whole expression becomes \( (a + b^{1/2})^{1/2} \).

Key Concepts

Square Roots

Square roots are one of the most common roots in mathematics. They are a type of radical that asks "what number, when multiplied by itself, will give the particular number under the root?". For example, the square root of 9 is 3 because when you multiply 3 by itself, you get 9. The symbol for square root is \( \sqrt{} \). So, \( \sqrt{9} = 3 \).
In problems involving square roots, you may encounter more complex terms like \( \sqrt{a + \sqrt{b}} \). This means you first resolve the innermost square root, then handle the outer one with regard to the context. Understanding square roots is essential for manipulating and simplifying expressions.

Exponent Notation

Exponent notation is a way to represent multiplication of the same number multiple times. In the context of roots, rational exponents are particularly useful. Rational exponents offer a more flexible method of expressing roots, including square roots. For instance, the square root of a number \( b \) is written as \( b^{1/2} \).
Exponent notation makes complex expressions easier to handle, allowing operations like multiplication or division to be carried out more smoothly. When you see an expression like \( \sqrt{b} \), you can convert it to exponent form so it reads \( b^{1/2} \). This opens the door to deeper manipulation, exploiting the laws of exponents for further simplification or expansion.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, operations, and sometimes functions, all put together in a meaningful manner. In the expression \( \sqrt{a + \sqrt{b}} \), each part has its own role. The letter \( a \) might be a constant or another expression part, while \( \sqrt{b} \) is an expression in its own right.
Handling mathematical expressions requires understanding each component and how they interact. With rational exponents, each element of the expression can be rewritten to reveal deeper insights. Transforming radicals into rational exponents, therefore, simplifies their integration into broader algebraic operations.

Algebraic Manipulation

Algebraic manipulation is the process of reshaping mathematical expressions using various techniques. This can involve distributing terms, factoring, or applying exponent rules. Given an expression like \( (a + b^{1/2})^{1/2} \), algebraic manipulation is essential for simplification or solving.
Key techniques include converting roots into exponents, as roots and exponents follow specific algebraic rules. By changing \( \sqrt{b} \) to \( b^{1/2} \), and consequently \( \sqrt{a + b^{1/2}} \) to \( (a + b^{1/2})^{1/2} \), we maintain the integrity of the expression but ease the way for further operations. Learning these manipulation strategies opens many pathways in both simple and complex algebraic contexts.