Problem 91 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[5]{x^{6} y^{2}}+\sqrt[5]{32 x^{6} y^{2}}+\sqrt[5]{x^{6} y^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 4x \cdot \sqrt[5]{x y^2} \).

Step 1: Identify Similar Terms

The expression is \( \sqrt[5]{x^{6} y^{2}} + \sqrt[5]{32 x^{6} y^{2}} + \sqrt[5]{x^{6} y^{2}} \). Notice that the first and the last terms are identical: \( \sqrt[5]{x^{6} y^{2}} \). We will combine these similar terms.

Step 2: Combine Like Terms

Since \( \sqrt[5]{x^{6} y^{2}} \) appears twice, add these two terms: \( \sqrt[5]{x^{6} y^{2}} + \sqrt[5]{x^{6} y^{2}} = 2 \cdot \sqrt[5]{x^{6} y^{2}} \). Now, the expression is \( 2 \cdot \sqrt[5]{x^{6} y^{2}} + \sqrt[5]{32 x^{6} y^{2}} \).

Step 3: Simplify Each Radical Term

Simplify each radical term by factoring out powers of the inner expression equal to the root. For \( \sqrt[5]{x^{6} y^{2}} \), \( x^{6} = (x^{5}) \cdot x \), giving us \( x \cdot \sqrt[5]{x y^2} \). For \( \sqrt[5]{32 x^{6} y^{2}} \), note that 32 is \( 2^5 \), so it simplifies to \( 2 \cdot x \cdot \sqrt[5]{x y^2} \).

Step 4: Substitute Simplified Forms Back into Expression

Substitute the simplified forms into the expression: \( 2 \cdot (x \cdot \sqrt[5]{x y^2}) + 2 \cdot x \cdot \sqrt[5]{x y^2} = 2x \cdot \sqrt[5]{x y^2} + 2x \cdot \sqrt[5]{x y^2} \).

Step 5: Combine All Simplified Terms

Since both terms are identical, combine them: \( 4x \cdot \sqrt[5]{x y^2} \).

Key Concepts

Combining Like Terms

Combining like terms is a foundational concept in algebra that helps simplify expressions by aggregating terms that have the same variables and exponents. In the given exercise, we have the expression

• \(\sqrt[5]{x^{6} y^{2}} + \sqrt[5]{32 x^{6} y^{2}} + \sqrt[5]{x^{6} y^{2}}\)

Observe that the first and last terms, \(\sqrt[5]{x^{6} y^{2}}\), are identical. When terms have similar variable components and roots, they can be combined by adding their coefficients. Here, the coefficient of \(\sqrt[5]{x^{6} y^{2}}\) before combination is understood as 1. Thus,

• 1 and 1 add up to give 2, resulting in the combined term \(2 \cdot \sqrt[5]{x^{6} y^{2}}\).

Combining terms reduces the complexity of equations and expressions, making them simpler to solve or manipulate further.

Simplifying Radicals

Simplifying radicals involves breaking down the expression under a root to its simplest form. Radicals often appear tricky, but they become manageable once factors are rearranged. Consider the radical \(\sqrt[5]{x^{6}y^2}\). Since \(x^{6}\) can be expressed as \((x^{5}) \cdot x\), the fifth root of \(x^{6}\) can be simplified by

• factoring out \(x^{5}\), which results in \(x \cdot \sqrt[5]{xy^2}\).

Similarly, for \(\sqrt[5]{32 x^{6} y^{2}}\), since 32 is equivalent to \(2^5\), it simplifies directly to

• \(2 \cdot \sqrt[5]{x^{6} y^{2}}\), which further reduces to \(2 \cdot x \cdot \sqrt[5]{xy^2}\).

Once radicals are simplified, they can be substituted back into the original equation, leading to easier operations in algebraic manipulations.

Exponents and Roots

Understanding exponents and roots is essential to simplifying expressions in algebra. Exponents indicate how many times a number is multiplied by itself, while roots are essentially the inverse, showing what number multiplied by itself gives the original number. In this exercise, the expression \(x^6\) under a fifth root was simplified by

• recognizing that \(x^6 = (x^5) \cdot x\), hence can be split into \(x \cdot \sqrt[5]{x}\).

The fifth root tells us to factor the number such that it fits inside the parameters of fivefold multiplication and division.

• Similarly, 32 as \(2^5\), perfectly fits the fifth root, getting rid of any root component in the simplification.

Combining knowledge of exponents and roots allows one to efficiently break down and reconstitute mathematical expressions, aiding in both manual computations and understanding complex algebraic concepts.