Problem 98 - Elementary and Intermediate Algebra

Question

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[8]{q^{2}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[8]{q^2} \) is \( q^{1/4} \).

Step 1: Understand the Radical Expression

The expression given is the eighth root of a power, \( \sqrt[8]{q^2} \). The goal is to simplify this expression using rational exponents.

Step 2: Convert the Radical to a Rational Exponent

Recall that the \( n \)-th root of a number \( a \) can be expressed using a rational exponent, specifically \( a^{1/n} \). For the given expression \( \sqrt[8]{q^2} \), this means we can write it as \( (q^2)^{1/8} \).

Step 3: Simplify the Rational Exponent

To simplify \( (q^2)^{1/8} \), we use the power of a power property: \( (a^m)^n = a^{m \cdot n} \). Thus, \((q^2)^{1/8} = q^{2 \times 1/8} = q^{2/8}\).

Step 4: Reduce the Fractional Exponent

Simplify the fraction in the exponent \( \frac{2}{8} \), which reduces to \( \frac{1}{4} \). Thus, the expression becomes \( q^{1/4} \). This is the simplified form of the original radical expression.

Key Concepts

Understanding Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are mathematical statements including a radical symbol (√) or its variations like the cube root (³√) or fourth root. In radicals, the number inside the root sign is called the radicand, and the small number outside, called the index, indicates the degree of the root, for example, an eighth root in \( \sqrt[8]{q^2} \).

• Radicand: The expression or number under the radical sign. In our example, it’s \(q^2\).
• Index: This is what determines the root. An eighth root means the radicand must be multiplied by itself eight times to revert to its original value.

Radical expressions can often be rewritten as expressions with fractional (rational) exponents, which can simplify calculation and manipulation. This is key when working to understand or simplify any radical expression.

Simplifying Expressions with Rational Exponents

Simplifying expressions typically involves reducing them to their simplest or most understandable form. When converting from a radical to an exponential form using rational exponents, simplification often follows. For instance, converting \( \sqrt[8]{q^2} \) into \( (q^2)^{1/8} \) uses the rational exponent format of radicals.

• Converting to rational exponents: The \( n\)-th root of an expression \( a \) becomes \( a^{1/n} \).
• Simplifying involves expressing the exponent in its simplest form, which may need prime factorization or finding common denominators.

In simplifying \( (q^2)^{1/8} \), reducing the exponent's fraction gives us \( q^{1/4} \). This fractional exponent form reflects the same relationship but is more manageable in mathematical operations, showing how exponent simplification makes expressions easier to work with.

Applying the Power of a Power Property

The power of a power property in exponents is a rule that simplifies the operation of exponents applied sequentially. Specifically, if an expression with an exponent is raised to another exponent, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

• This property is invaluable when handling nested powers, allowing effective simplification.
• Look for scenarios where applying this property can reduce complex expressions.

In our example, \((q^2)^{1/8}\), the power property simplifies to \( q^{2 \times 1/8} = q^{2/8} \) which is further simplified using basic fraction reduction. This step is crucial in transforming and simplifying expressions, showing the power of combining exponent rules.