Problem 117 - Intermediate Algebra

Question

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. $$ \sqrt[5]{x^{35}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^{7} \).

Step 1: Review Expression

The expression given is \( \sqrt[5]{x^{35}} \). It is important to know that when dealing with roots and exponents, we can apply the property of exponents \( (a^{m})^{n} = a^{mn} \).

Step 2: Apply Exponent Property to Simplify

Re-write the 5th root in terms of fractional exponents: \( x^{35} \) under the 5th root is equivalent to \( x^{\frac{35}{5}} \).

Step 3: Simplify the Exponent

Calculate the simplified exponent by doing the division in the fractional exponent: \( \frac{35}{5} = 7 \).

Step 4: Write Final Simplified Expression

After simplifying the exponent, the expression \( x^{\frac{35}{5}} \) becomes \( x^{7} \). Thus, the simplified expression is \( x^{7} \).

Key Concepts

Fractional Exponents

Fractional exponents are a way to express powers and roots together in one notation. For instance, the expression \( x^{\frac{a}{b}} \) means that we are taking the \( b \)th root of \( x \) and then raising the result to the power of \( a \). Understanding fractional exponents allows us to translate roots into a more unified form that works seamlessly with the general rules of exponents.

Here’s a quick breakdown of why fractional exponents are useful:

• They simplify the process of dealing with complex roots.
• They allow for easier manipulation of equations involving both roots and exponents.
• They provide a consistent way to apply exponential rules.

In the provided exercise, expressing the 5th root of \( x^{35} \) with fractional exponents gives us \( x^{\frac{35}{5}} \). This form allows for straightforward calculation and simplification.

Radicals and Roots

Radicals represent roots, such as square roots or cube roots, and are expressed using a radical sign (\( \sqrt{} \)). The number outside the radical sign indicates the degree of the root. For example, \( \sqrt[5]{x} \) means the 5th root of \( x \).

Understanding radicals and roots is essential because they often appear in algebra and geometry, where we deal with both whole numbers and variables. Here are some key points about radicals:

• Roots can be expressed as fractional exponents (e.g., \( \sqrt[5]{x} = x^{\frac{1}{5}} \)).
• The process of simplifying radicals involves expressing them in their simplest form by removing any possible perfect powers.
• Radicals can denote the root of both numerical constants and algebraic expressions.

In our exercise, the radical \( \sqrt[5]{x^{35}} \) indicates that we are looking for the 5th root of \( x^{35} \), which we simplify using the properties of fractional exponents.

Simplifying Expressions

Simplifying expressions often involves reducing them to their most manageable form. This makes them easier to interpret and use in subsequent calculations. Here are some general strategies involved in simplifying expressions:

• Combine like terms, especially when dealing with polynomial expressions.
• Use exponent rules to condense or expand expressions.
• Convert roots and radicals into fractional exponents for easier manipulation.
• Simplify fractions by finding common denominators or reducing to lowest terms.

In our specific problem, we began with \( \sqrt[5]{x^{35}} \), an expression that seems quite complex initially. By re-writing the root as a fractional exponent, \( x^{\frac{35}{5}} \), we can simplify the exponent by performing simple division, resulting in a much simpler expression, \( x^{7} \). Simplifying in this way not only cleans up the expression but also clarifies the underlying mathematics.