Problem 498 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{64 a^{10}}-\sqrt[3]{-216 a^{12}}\) (b) \(\sqrt[4]{486 u^{7}}+\sqrt[4]{768 u^{3}}\)

Step-by-Step Solution

Answer

(a) 4 a^{10/3} - 6 a^4(b) 3 \sqrt[4]{6} u^{7/4} + 2 \sqrt[4]{3 u^{3}}

Step 1: Understand Each Radical Expression

First, recognize the expressions inside the radicals. For part (a), these are \( \sqrt[3]{64 a^{10}} \) and \( \sqrt[3]{-216 a^{12}} \). For part (b), these are \( \sqrt[4]{486 u^{7}} \) and \( \sqrt[4]{768 u^{3}} \).

Step 2: Simplify the Cube Roots in Part (a)

For cube roots, express the radicand in terms of perfect cubes. Simplify \( \sqrt[3]{64 a^{10}} \). Since \( 64 = 4^3 \) and \( 64 a^{10} = (4 a^3)^{3/3} \times a \), we get \( \sqrt[3]{64 a^{10}} = 4 a^3 \times a = 4 a^{10/3} \). For the second term, \( \sqrt[3]{-216 a^{12}} = \sqrt[3]{-6^3 a^12} = -6 a^4 \). Now we have \( 4 a^{10/3} - 6 a^4 \).

Step 3: Simplify the Fourth Roots in Part (b)

For fourth roots, express the radicand in terms of perfect fourth powers. Simplify \( \sqrt[4]{486 u^{7}} \). \( 486 = 3^5 \times 2 \). We can write \( 486 u^{7} \) as \( (3^3 \times (3 \times 2) u^4) u^3 \), which results in \( 3 \times \sqrt[4]{6} u^{3/2} \). Next, simplify \( \sqrt[4]{768 u^{3}} \). \( 768 = 2^8 \times 3 \), which gives us \( \sqrt[4]{768 u^{3}} = 2^{2} \sqrt[4]{3 u^{3/4}} = 2 \sqrt[4]{3 u^{3}} \).

Step 4: Combine and Simplify Both Parts

For (a), combine: \( 4 a^{10/3} - 6 a^4 \). For (b), combine: \( 3 \sqrt[4]{6} u^{7/4} + 2 \sqrt[4]{3 u^3} \).

Key Concepts

Cube Roots

Understanding cube roots is crucial for simplifying radical expressions, especially in the given problem where we encounter \[ \sqrt[3]{64 a^{10}} \] and \[- \sqrt[3]{-216 a^{12}} \]. Cube roots involve finding what number, when multiplied by itself three times, gives the radicand. In mathematical terms, if \(\root[3]{x} = y\), then \((y \times y \times y = x)\).
To simplify, break down the radicand into prime factors and reorganize it to identify the perfect cubes. For example:

• For \[ \sqrt[3]{64 a^{10}}\] we recognize that \[64 = 4^3\] and we separate \[a^{10}\] as \[a^{9} * a\], meaning \[64 a^{10} = (4 a^3)^3 \times a \]. Thus, the cube root becomes \[4 a^3 \times a = 4 a^{10/3}\].
• For \[ \sqrt[3]{-216 a^{12}}\] note that \[-216 = -6^3\] and \[a^{12} = a^{4 \times 3}\], therefore, the expression simplifies to \[-6 a^4\].

Fourth Roots

Fourth roots require finding what number, when multiplied by itself four times, results in the radicand. Mathematically, if \(\root[4]{x} = y\), then \((y \times y \times y \times y = x)\). For the problem, we look at terms like \[ \sqrt[4]{486 u^{7}}\] and \[ \sqrt[4]{768 u^{3}}\]. Rewriting these:

• For \[ \sqrt[4]{486 u^{7}}\]: Rewrite \[486 as 3^5 \times 2\], then express the radicand \[486 u^7 = (3^4 \times 3 \times 2 \times u^4) u^3\], resulting in \[3 \times \sqrt[4]{6} u^{7/4} or 3\root[4]{6} u^{7/4}\].
• For \[\sqrt[4]{768 u^{3}}\]: Rewrite \[768 as 2^8 \times 3\], which becomes \[\sqrt[4]{768 u^3} = 2^2 \sqrt[4]{3 u^{3}} = 2 \sqrt[4]{3 u^{3}}\].

Identifying the perfect fourth powers and simplifying will help reduce the expressions efficiently.

Radicands

A radicand is the number or expression under the radical sign (like inside \[ \sqrt{} \] or \[\sqrt[3]{} \]). Understanding the composition of a radicand helps in simplifying radical expressions. For instance, in expressions like \[ \sqrt[3]{64 a^{10}}\], the radicand is \[64 a^{10}\]. This radicand determines how we simplify the expression.
For simplifying:

• First identify any perfect powers within the radicand. In \[64 ~ a^{10}\], we know \[64 = 4^3\] and \[a^{10} = (a^3)^3 \times a\].
• Re-express the radicand in these terms to simplify: \[ 64 a^{10} = (4 a^3)^3 \times a\], which simplifies to \[4 a^{10/3}\].
• For fourth roots, in \[486 u^{7}\], first express 486: \[486 = 3^5 \times 2\], thus \[486 u^{7} = (3^3 \times (3 \times 2) u^4) u^3\], leading to \[3 \times \sqrt[4]{6} u^{7/4}\].

Recognizing and manipulating these radicands assists greatly in obtaining the simplified forms.

Simplification

Simplification is a critical step in manipulating radical expressions to their simplest form. It involves breaking down complex expressions into more manageable ones.
This often includes:

• Identifying and extracting perfect powers within the radicand.
• Expressing the radical in terms of these perfect powers for easier root calculation.

Consider the exercise example:

• For \[\sqrt[3]{64 a^{10}}\]: Simplify using \[\sqrt[3]{64}\] and \[\sqrt[3]{a^{10}}\] separately.
• Combining gives \[4 a^{10/3}\] because \[4\] is the cube root of \[64\] and \[a^{10} = a^{3+3+3+1 (a^3 * a^3 * a^3 * a) \rightarrow a^1 (4 times the multiplication of a = 3times accountability a^3\}\ of\].
• Repeating this process for other parts of the problem: \[\sqrt[3]{-216 a^{12}} = -6 a^4\], simplifies to \[-6 a^4\].
• Combine expression and simplifying further for the ultimate solution: \[\root[4]{486 u^7} = 3 \sqrt[4]{6} u^{7/4}\] and for involving number 768 \sqrt[4]{(2^8 * 3)} u^3= 2^2 \[Position\].

These simplification steps will ensure that you arrive at the minimal and most elegant form of any radical expression.