Problem 73 - Algebra and Trigonometry

Question

Simplify the radical expressions in Exercises \(67-74\) if possible. $$ \frac{\sqrt[5]{64 x^{6}}}{\sqrt[5]{2 x}} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{5}{3}x \).

Step 1: Break down the expression

Firstly, rewrite the given expression as \( \frac{(64x^6)^{1/5}}{(2x)^{1/5}} \).

Step 2: Simplify using the laws of exponents

Use the rule that a fraction raised to an exponent is equivalent to the numerator and the denominator raised separately to that same exponent. This gives us \( \frac{64^{1/5}*(x^6)^{1/5}}{2^{1/5}*x^{1/5}} \).

Step 3: Calculate the fifth root

Calculate the fifth root for \$64\$ and \$2\$, which results in \(2\) and \(1.2\) respectively. Thus, obtaining \( \frac{2*(x^6)^{1/5}}{1.2*x^{1/5}} \).

Step 4: Simplify exponents

Next is to simplify \((x^6)^{1/5}\) and \(x^{1/5}\) separately using the rule of powers, which gives \( \frac{2*x^{6/5}}{1.2*x^{1/5}} \).

Step 5: Reduce terms

Simplify by reducing terms that appear both in the numerator and denominator which results in \(\frac{2}{1.2}*x^{(6-1)/5} \Rightarrow \frac{5}{3}*x^1 = \frac{5}{3}x\) after further simplification.

Key Concepts

Laws of Exponents

Understanding the Laws of Exponents is crucial for simplifying radical expressions. When dealing with exponent rules, there are several key principles to keep in mind:

• Product Rule: When multiplying two exponents with the same base, you add the exponents, as in: \( a^m \times a^n = a^{m+n} \).
• Quotient Rule: When dividing two exponents with the same base, subtract the exponents: \( a^m \/ a^n = a^{m-n} \).
• Power Rule: When raising an exponent to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Root Rule: Taking the root of an exponent is the inverse of raising it to a power, which is essentially dividing the exponent by the root: \( \sqrt[n]{a^m} = a^{m/n} \).

These rules can be used in various combinations to simplify complex expressions. By employing the laws correctly, the simplification process becomes systematic and manageable. For instance, when faced with a fraction, the Quotient Rule is particularly handy as it helps to reduce terms, casting them in a simpler form as seen in the exercise solution above, where exponents are subtracted during the simplification process.

Fifth Root Calculation

The fifth root of a number is one of its five equal factors. In mathematical terms, finding the fifth root is the inverse operation of raising a number to the fifth power. For example, the fifth root of 32 is 2 because \( 2^5 = 32 \).

When calculating the fifth root, which is notated as \( \sqrt[5]{...} \), you're looking for a number that, when multiplied by itself four more times, will equal the original number inside the radical. Calculations of perfect fifth roots, such as with the example of \( \sqrt[5]{32} = 2 \), are straightforward; however, with non-perfect powers, the calculation may not result in a whole number, and an approximation may be necessary. In the provided exercise, to simplify the expression, a fifth root calculation is applied to both numbers and variables. The fifth root of a variable to a power, like \( \sqrt[5]{x^5} \), would simplify to the variable itself, since \( x^{5/5} = x^1 = x \).

Simplify Exponents

Simplifying exponents is a matter of applying the rules of exponents effectively to an algebraic expression. One common technique is to 'break down' the expression into smaller parts, as we saw in the step by step solution where \( (x^6)^{1/5} \) is simplified to \( x^{6/5} \) by applying the Power Rule.

Another key aspect to simplify exponents is to reduce any similar terms between the numerator and the denominator. For instance, if we have \( x^n \) in the numerator and \( x^m \) in the denominator, and if \( n > m \), we can simplify the expression to \( x^{n-m} \) by cancelling out the common factor using the Quotient Rule.

In the given exercise, after applying the fifth root calculation, the power of \( x \) was simplified by subtracting exponents in the numerator and denominator. The expression \( x^{6/5} \) over \( x^{1/5} \) was simplified by applying the Quotient Rule, which resulted in \( x^{(6-1)/5} = x^1 = x \), further reducing the expression to its simplest form.