Problem 471 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[5]{96 a^{7}}\) (b) \(\sqrt[3]{375 b^{4}}\)

Step-by-Step Solution

Answer

Simplified (a): \( 2a^{7/5} \sqrt[5]{3} \). Simplified (b): \( 5b^{4/3} \sqrt[3]{3} \).

Step 1: Understand Fifth Root and Simplify

Begin with \(\sqrt[5]{96 a^{7}}\). We need to break down 96 and the variable part separately. Start by simplifying 96: \(96 = 2^5 \times 3\). Rewrite the expression inside the root: \(\sqrt[5]{2^5 \times 3 \times a^7}\).

Step 2: Apply the Fifth Root

Apply the fifth root to each part: \(\sqrt[5]{2^5} \times \sqrt[5]{3} \times \sqrt[5]{a^7}\). This becomes: \(2 \times \sqrt[5]{3} \times a^{7/5}\).

Step 3: Simplify the Expression

Combine the simplified terms to get: \(2a^{7/5} \sqrt[5]{3}\). Now the expression is simplified.

Step 4: Understand Cube Root and Simplify

Now simplify \( \sqrt[3]{375b^4} \). Begin by breaking down 375: \( 375 = 3 \times 5^3 \). Rewrite the expression inside the root: \( \sqrt[3]{3 \times 5^3 \times b^4} \).

Step 5: Apply the Cube Root

Apply the cube root to each part: \( \sqrt[3]{3} \times \sqrt[3]{5^3} \times \sqrt[3]{b^4} \). This becomes: \( \sqrt[3]{3} \times 5 \times b^{4/3} \).

Step 6: Simplify the Expression

Combine the simplified terms to get: \( 5b^{4/3} \sqrt[3]{3} \). Now the expression is simplified.

Key Concepts

Understanding the Fifth Root

A fifth root is the number that, when raised to the power of five, gives the original number. It is expressed as \(\backslashsqrt[5]{x}\). For example, the fifth root of 32 is 2, because \2^5 = 32\. To simplify a fifth root expression, you look to separate it into its prime factors. This helps you to easily identify which parts can be extracted from the radical. If a number inside the fifth root can be expressed as a factor raised to the power of 5, that factor can be moved outside the radical easily.

Breaking Down the Cube Root

Just like the fifth root, a cube root is the number that, when multiplied by itself three times, gives the original number. It is represented as \(\backslashsqrt[3]{x}\). Understanding how to find the cube root is crucial in simplifying radical expressions. For instance, the cube root of 27 is 3 because \3^3 = 27\. Similar to fifth roots, you’ll separate the number inside the cube root into its prime factors. Any factor that appears three times can be simplified by bringing one instance of that factor outside the cube root.

Mastering Exponentiation

Exponentiation involves raising a number (the base) to a power (the exponent). For example, \a^5\ means multiplying \a\ by itself five times. This concept also helps in understanding roots. For example, the fifth root of \a^7\ can be rewritten as \(a^{7/5}\). This shows that finding roots is the inverse operation of exponentiation. Remember that the exponent in the root becomes the denominator in a fractional exponent. For example, the cube root of \b^4\ translates to \(b^{4/3}\).

Simplification Steps

Simplifying radical expressions involves a series of systematic steps.
1. **Break Down the Number:** Start by breaking down the number inside the radical into its prime factors.
2. **Separate Variables and Constants:** Treat the number and variable parts independently.
3. **Apply the Root:** Apply the root to both the numerical and variable parts. For example, \(\backslashsqrt[5]{96 a^7}\) becomes \(\backslashsqrt[5]{96} \backslashsqrt[5]{a^7}\).
4. **Simplify Each Part:** Extract the factors that match the root from inside the radical. For instance, \2^5\ in fifth root becomes \2\ outside the root. Variables like \a^7\ under a fifth root become \a^{7/5}\.
5. **Combine Terms:** Finally, bring together the simplified parts. For \(\backslashsqrt[3]{375 b^4}\), you get \5b^{4/3} \backslashsqrt[3]{3}\.