Problem 82 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \sqrt{s \sqrt{s}^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(s^{\frac{3}{4}}\).

Step 1: Express the square root terms in exponential form

The square root of a number can be expressed as that number raised to the power of \(\frac{1}{2}\). Similarly, if you have \(\sqrt{s^3}\), it can be written as \((s^3)^{\frac{1}{2}}\).

Step 2: Simplify the inner exponentiation

Apply the rule \((a^m)^n = a^{m \cdot n}\) to the expression \((s^3)^{\frac{1}{2}}\). This becomes \(s^{3 \cdot \frac{1}{2}} = s^{\frac{3}{2}}\).

Step 3: Simplify the outer square root

Now we have the expression \(\sqrt{s^{\frac{3}{2}}}\). Express this as \((s^{\frac{3}{2}})^{\frac{1}{2}}\). Applying the same rule \((a^m)^n = a^{m \cdot n}\), this becomes \(s^{\frac{3}{2} \cdot \frac{1}{2}} = s^{\frac{3}{4}}\).

Step 4: Express the final result using rational exponents

The simplified expression, using rational exponents, is \(s^{\frac{3}{4}}\).

Key Concepts

Exponential Form

Understanding exponential form is key when dealing with expressions that involve roots and powers. Exponential form means representing roots as powers of fractions. For the square root of a number, it is expressed as raising the number to the power of \(\frac{1}{2}\). This helps to simplify complex expressions by using the properties of exponents. In the given exercise, the expression \(\sqrt{s \sqrt{s}^3}\) uses exponential form to simplify the nested roots. The square root \(\sqrt{s^3}\) becomes \((s^3)^{\frac{1}{2}}\), which allows further simplification by manipulating exponents. Expressing roots in exponential form is powerful because it allows us to leverage the known rules of exponents for simplification.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise form without changing their value. In algebra, this often involves transforming roots into exponents to make use of exponent rules. For instance, when we have \(\sqrt{s^3}\), simplifying means applying the exponent rule \((a^m)^n = a^{m \cdot n}\). This converts \(s^3\) to \((s^3)^{\frac{1}{2}} = s^{\frac{3}{2}}\).

Next, we simplify the expression further: \(\sqrt{s^{\frac{3}{2}}}\) is rewritten as \((s^{\frac{3}{2}})^{\frac{1}{2}}\), which simplifies to \(s^{\frac{3}{4}}\) using the same exponent rule. By expressing everything in terms of exponents, we streamlined the expression into a single power of \(s\), making the whole process clearer and the result easier to interpret.

Fundamental Algebra Concepts

To master algebraic simplification, grasp fundamental concepts like the properties of exponents and roots. One key property used here is \((a^m)^n = a^{m \cdot n}\), which allows us to handle expressions involving both powers and roots effectively.

Another fundamental idea is that any root can be expressed as a fractional exponent, which unifies the treatment of roots within algebraic expressions. This simplification aids in performing operations such as multiplication or division of powers directly.
By converting roots to exponents, we leverage these properties, streamlining calculations and reducing the complexity of algebraic manipulations. Thus, mastering these core concepts not only simplifies expressions but also provides the tools needed to tackle more complex algebraic challenges.