Problem 53 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \frac{w^{4 / 3} w^{2 / 3}}{w^{1 / 3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( w^{5/3} \).

Step 1: Simplify the Numerator

To simplify the numerator of the expression \( w^{4/3} w^{2/3} \), we apply the rule of exponents which states that when multiplying like bases, we add the exponents. Thus, \( w^{4/3} \times w^{2/3} = w^{(4/3 + 2/3)} = w^{6/3} \).

Step 2: Simplify the Exponent

Now, simplify \( w^{6/3} \). This simplifies to \( w^2 \) since dividing 6 by 3 yields 2. Thus, the numerator becomes \( w^2 \).

Step 3: Simplify the Whole Expression

Replace the simplified numerator back into the expression, giving \( \frac{w^2}{w^{1/3}} \). To divide powers of the same base, subtract the exponent in the denominator from the exponent in the numerator: \( w^{2 - 1/3} \).

Step 4: Calculate the Exponent

Subtract the exponent \(1/3\) from 2: \[ 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3} \]. So, \( w^{2 - 1/3} = w^{5/3} \).

Step 5: Final Answer

The expression simplifies to \( w^{5/3} \), and it already has positive exponents, so no further simplification is needed.

Key Concepts

Exponent Rules

Exponent rules are the basic principles that guide us in simplifying expressions involving powers.
In mathematics, an exponent refers to how many times a number, known as the base, is multiplied by itself. Here are a few important rules to remember when working with exponents:

• Product of Powers Rule: When you multiply two exponents with the same base, you add their exponents. For example, \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing two exponents with the same base, subtract the exponent of the denominator from that of the numerator. Example: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For instance, \( (a^m)^n = a^{m \times n} \).

Understanding these rules helps to simplify complex expressions quickly and accurately, as shown in the step-by-step solution.

Negative Exponents

Negative exponents can be a bit tricky, but they are easier than they seem. A negative exponent indicates that the base is on the wrong side of a fraction and needs to be flipped.

For instance, when you see \( a^{-n} \), this can be rewritten as \( \frac{1}{a^n} \). So, when you encounter a negative exponent, simply take the reciprocal of the base raised to the positive of that exponent. Here are additional points to consider:

• Moving a factor across the fraction line changes the sign of its exponent.
• Always aim to have positive exponents in your final answer for simplicity and clarity.

In our given solution, we didn't have negative exponents, but knowing how to handle them is crucial for algebraic simplification.

Algebraic Simplification

Algebraic simplification involves making expressions easier to work with or solve.
This often means reducing an expression to its simplest form by using mathematical rules and properties. In the original exercise, algebraic simplification involves using exponent rules to combine and reduce terms:

• First, simplify the numerator by using the product of powers rule: \( w^{4/3} \times w^{2/3} = w^{6/3} \).
• Then, simplify \( w^{6/3} \) to \( w^2 \) because \( 6/3 \) equals 2.
• Substitute this simplified form back into the expression, \( \frac{w^2}{w^{1/3}} \), and apply the quotient of powers rule to get \( w^{5/3} \).

By simplifying, you can more easily evaluate or solve expressions, making complex problems much more manageable. The goal of simplification is to get clear and concise expressions for efficient problem-solving.