Problem 57 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[8]{\sqrt[4]{a} \cdot \sqrt[4]{b^{7}}} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \( a^{\frac{1}{32}} \cdot b^{\frac{7}{32}} \).

Step 1: Convert Inner Radicals to Powers

Begin by expressing the inner radicals as powers. The expression \( \sqrt[4]{a} \) is equivalent to \( a^{\frac{1}{4}} \). Similarly, \( \sqrt[4]{b^{7}} \) can be written as \( b^{7 \times \frac{1}{4}} = b^{\frac{7}{4}} \). The expression under the outer radical is now \( a^{\frac{1}{4}} \cdot b^{\frac{7}{4}} \).

Step 2: Combine Powers Inside the Radical

Using the properties of exponents, combine the powers of \( a \) and \( b \) together: \( (a^{\frac{1}{4}} \cdot b^{\frac{7}{4}}) = a^{\frac{1}{4}}b^{\frac{7}{4}} \). The expression for the inner radical remains the same as \( \sqrt[8]{a^{\frac{1}{4}}b^{\frac{7}{4}}} \).

Step 3: Apply the Outer Radical

Convert the outer radical to a power expression: \( \sqrt[8]{a^{\frac{1}{4}}b^{\frac{7}{4}}} \) can be rewritten using the radicand's exponent as \( (a^{\frac{1}{4}}b^{\frac{7}{4}})^{\frac{1}{8}} \).

Step 4: Simplify the Expression by Applying the Power Rule

Use the power of a power rule: \( (x^m)^n = x^{m \cdot n} \). Therefore, \( (a^{\frac{1}{4}})^{\frac{1}{8}} = a^{(\frac{1}{4}) \cdot (\frac{1}{8})} = a^{\frac{1}{32}} \) and \( (b^{\frac{7}{4}})^{\frac{1}{8}} = b^{(\frac{7}{4}) \cdot (\frac{1}{8})} = b^{\frac{7}{32}} \).

Step 5: Final Simplification to Achieve the Simplest Form

Combine the simplified powers to express the radical in its simplest form: \( a^{\frac{1}{32}} \cdot b^{\frac{7}{32}} \).

Key Concepts

Properties of Exponents

Exponents are an essential part of algebra that help us express repeated multiplication compactly. The properties of exponents simplify working with large numbers or complex expressions.
Understanding these properties is crucial as they allow us to manipulate expressions involving powers easily. Here are some key properties:

• Product of Powers: When multiplying like bases, the exponents are added: \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product: When raising a product to a power, distribute the exponent to each factor: \( (ab)^m = a^m \cdot b^m \).
• Quotient of Powers: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \) (\(a eq 0\)).

These properties provide the foundation for simplifying expressions and are especially helpful when dealing with radical expressions, as seen in many algebraic scenarios.

Simplifying Radicals

The process of simplifying radicals involves expressing roots in their simplest form, which often makes it easier to comprehend and solve problems. Simplifying radical expressions helps us to work with complex numbers more efficiently.
Radicals are often converted to fractional exponents to simplify them. For instance, the square root \( \sqrt{a} \) can be expressed as \( a^{1/2} \), while the cube root is \( a^{1/3} \). This conversion is foundational when working with nested radicals or when different roots are involved.

To simplify, follow these steps:

• Convert any radical to an exponent expression using fractional exponents.
• Apply the properties of exponents to combine and reduce terms.
• Re-express the result if necessary, in either radical or exponential form.

This process is crucial for solving equations and to provide answers in a clearer, simplified form.

Power of a Power Rule

The power of a power rule is pivotal when dealing with exponents. It makes simplifying nested exponents and expressions straightforward. The rule states that if you have an exponent raised to another exponent, you multiply the exponents together.
For example, \( (x^m)^n = x^{m \cdot n} \). This rule is widely used in various mathematical scenarios, including solving complex problems involving radical expressions.
The application of this rule helps with:

• Transforming nested radical expressions into a more manageable form.
• Simplifying calculations involving multiples of geometric progressions.
• Reducing the complexity of algebraic equations.

Applying the power of a power rule ensures our expressions remain easily interpretable, especially when multiple layers of exponents are involved.

Positive Exponents

Working with positive exponents helps in clearly expressing the value of a number or variable. When exponents are positive, they indicate a straightforward multiplication process. Positive exponents ensure clarity and facilitate easy comparison of numbers.
In algebra, expressing results using positive exponents makes them easier to understand and calculate further. This approach is particularly useful when dealing with radical expressions, as it eliminates the complexity associated with negative exponents.

For instance, converting a radical expression into a form with positive exponents involves:

• Expressing all factors with exponents.
• Adjusting any negative exponents using the inverse property.
• Combining like terms using the properties of exponents.

The use of positive exponents results in a much clearer, simplified form suitable for mathematical operations, including addition, subtraction, and comparison.