Problem 62 - Intermediate Algebra

Question

Multiply. $$ \frac{\left(2 x^{1 / 5}\right)^{4}}{x^{3 / 10}} $$

Step-by-Step Solution

Answer

The result is \( 16\sqrt{x} \).

Step 1: Simplify the Numerator

First, deal with the expression in the numerator. You need to simplify \( (2x^{1/5})^4 \). This involves distributing the exponent across both the coefficient and the variable inside the parenthesis:\[ (2x^{1/5})^4 = 2^4 \times (x^{1/5})^4 = 16 \times x^{(1/5) \times 4} = 16x^{4/5} \]

Step 2: Write the Entire Fraction

Now that you've simplified the numerator, write the entire fraction again using this simplified form:\[ \frac{16x^{4/5}}{x^{3/10}} \]

Step 3: Simplify the Fraction Using Laws of Exponents

To simplify the fraction, you need to use the laws of exponents. When you divide two expressions with the same base, you subtract the exponents:\[ x^{4/5 - 3/10} \]

Step 4: Subtract the Exponents

To subtract the exponents \( \frac{4}{5} \) and \( \frac{3}{10} \), you need to find a common denominator. The least common denominator of 5 and 10 is 10. Rewriting the exponents gives:\[ x^{\frac{8}{10} - \frac{3}{10}} = x^{\frac{5}{10}} = x^{1/2} \]

Step 5. Write Final Answer

Include the coefficient from the numerator to write the final answer:\[ 16x^{1/2} = 16\sqrt{x} \]

Key Concepts

Laws of Exponents

When working with exponents, there are some rules we call the "laws of exponents" that help us perform calculations more easily. These laws apply when we're working with the same base, like the variable \( x \). Here are a few crucial points to remember:

• **Multiplication of Like Bases:** When you multiply expressions with the same base, you **add** the exponents. For example, \( x^a \times x^b = x^{a+b} \).
• **Division of Like Bases:** When dividing, you **subtract** the exponents: \( \frac{x^a}{x^b} = x^{a-b} \).
• **Power of a Power:** When you raise a power to another power, multiply the exponents: \( (x^a)^b = x^{a \times b} \).

Understanding these principles not only simplifies calculations but also builds a strong foundation for tackling more complex algebraic problems.

Simplifying Expressions

Simplifying expressions, particularly those that involve exponents, is about following a series of logical steps to make an expression easier to work with. Let's consider the expression \((2x^{1/5})^4\).

• **Distribute the Exponent:** Apply the exponent to both the coefficient (2) and the variable part \( x^{1/5} \).

This gives you \( 2^4 \times (x^{1/5})^4 \). Break it down step-by-step:

• Calculate \( 2^4 \), which equals 16.
• For \((x^{1/5})^4\), multiply the exponents: \( x^{(1/5) \times 4} = x^{4/5} \).

This method of simplification leads us to 16\( x^{4/5} \), a simpler and cleaner form. Whenever you simplify, always make sure you apply the correct rules of exponents to avoid mistakes.

Fractional Exponents

Fractional exponents can be intimidating, but they are quite manageable if broken down. A fractional exponent, such as \( x^{1/5} \), indicates both a root and a power:

• **The Numerator (Power):** Tells you the power to which the base is raised.
• **The Denominator (Root):** Indicates the root being taken. For \( x^{1/5} \), this means the fifth root.

When multiplying or dividing expressions with fractional exponents, the key is to find a common base and then combine or simplify accordingly. For example, subtracting fractional exponents requires finding a common denominator, much like handling fractions. In our exercise, to simplify \( x^{4/5} \) by dividing it by \( x^{3/10} \), convert \( 4/5 \) to \( 8/10 \) and carry out subtraction: \( \frac{8}{10} - \frac{3}{10} = \frac{5}{10} = \frac{1}{2} \). This leads you to the simplified expression \( x^{1/2} \), which is equivalent to \( \sqrt{x} \). Thus, fractional exponents allow you to seamlessly transition between roots and powers.