Problem 482 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\frac{\sqrt[4]{64}}{\sqrt[4]{2}}\) (b) \(\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}\)

Step-by-Step Solution

Answer

(a) \( \root{4}{32} \). (b) \( \root{5}{64} x^{6/5} \).

Step 1: Simplify the Expression for (a)

Start by simplifying the expression \(\frac{\root{4}{64}}{\root{4}{2}}\). Rewrite the fourth roots as exponents: \(\frac{64^{1/4}}{2^{1/4}}\).

Step 2: Division of Exponents in (a)

Recall the rule for division of exponents with the same base: \(a^{m} / a^{n} = a^{m-n}\). Apply this rule: \({64^{1/4}} / {2^{1/4}} = (64 / 2)^{1/4} = 32^{1/4}\).

Step 3: Simplification of (a)

Recognize that \(\root{4}{32}\) isn't a perfect fourth root, so leave the expression as it is: \(\root{4}{32}\).

Step 4: Simplify the Expression for (b)

Rewrite the fifth roots in the fraction: \(\frac{\root{5}{128 x^{8}}}{\root{5}{2 x^{2}}} \rightarrow \frac{(128 x^{8})^{1/5}}{(2 x^{2})^{1/5}}\).

Step 5: Division of Exponents in (b)

Use the exponent rule for division: \(a^{m} / a^{n} = a^{m-n}\). Apply this rule: \(\frac{128^{1/5} \times x^{8/5}}{2^{1/5} \times x^{2/5}} = \frac{128^{1/5}}{2^{1/5}} \times \frac{x^{8/5}}{x^{2/5}}\).

Step 6: Further Simplification in (b)

Simplify \(128^{1/5} / 2^{1/5}\) to get \(64^{1/5}\), and \(x^{8/5} / x^{2/5} = x^{(8/5 - 2/5)} = x^{6/5}\). The final expression is \(\root{5}{64} \times x^{6/5}\).

Key Concepts

Fourth Roots

A fourth root of a number is a value that, when multiplied by itself four times, gives the original number. For example, the fourth root of 16 is 2, because \(2 \times 2 \times 2 \times 2 = 16\). Fourth roots are helpful in various algebraic simplifications when dealing with radical expressions. If you see \(\root{4}{a}\), you can also express it as \(a^{1/4}\). This form is often more convenient for applying exponent rules.

Exponent Rules

Exponent rules are essential when simplifying expressions involving powers. Here are a few key rules:

• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of powers: \(a^m / a^n = a^{m-n}\)
• Power of a power: \( (a^m)^n = a^{mn} \)

These rules help simplify complex expressions. For instance, simplifying \( \frac{64^{1/4}}{2^{1/4}} \) involves subtracting exponents since the bases (4th roots in this case) are the same.

Fifth Roots

A fifth root of a number is a value that, when multiplied by itself five times, gives the original number. For example, the fifth root of 32 is 2, because \(2 \times 2 \times 2 \times 2 \times 2 = 32\). Fifth roots are denoted as \(\root[]{5}{a}\) or \(a^{1/5}\). This notation is very important in higher-level algebra, especially when dealing with polynomial equations and radical expressions.

Fractional Exponents

Fractional exponents are another way of representing roots. For example, \(a^{1/2} = \root[]{2}{a} \) (which is the square root), and \(a^{1/3} = \root[]{3}{a} \) (which is the cube root). These are extremely useful for simplifying expressions. When you see \(a^{m/n}\), this means you first take the nth root of a, and then raise it to the mth power (or vice versa). This makes calculations easier and often simpler when using exponent rules.

Simplification in Algebra

Simplification in algebra involves reducing an expression to its simplest form. Key steps include:

• Combining like terms
• Applying exponent rules
• Factoring
• Cancelling common factors

For example, simplifying \( \frac{(128 x^8)^{1/5}}{(2 x^2)^{1/5}} \) involves using exponent rules to separate the numbers and variables and then reduce each part step by step. Simplification not only makes expressions easier to work with but also helps in solving equations more efficiently.