Problem 39 - Algebra A Combined Function

Question

Write with positive exponents. Simplify if possible. $$ \frac{5}{7 x^{-3 / 4}} $$

Step-by-Step Solution

Answer

\( \frac{5x^{3/4}}{7} \)

Step 1: Write the Expression with Positive Exponents

The original expression is \( \frac{5}{7 x^{-3/4}} \). We need to eliminate the negative exponent. In mathematics, \( x^{-a} = \frac{1}{x^a} \). So, \( x^{-3/4} = \frac{1}{x^{3/4}} \). Substitute this into the expression to get: \[ \frac{5}{7 \cdot \frac{1}{x^{3/4}}} = \frac{5 \cdot x^{3/4}}{7}. \]

Step 2: Simplify the Expression

Now we have \( \frac{5 \cdot x^{3/4}}{7} \). This expression is already written with positive exponents. Since there are no further opportunities to simplify the numeric fraction or combine terms, this is the simplest form of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making them easier to understand and use without changing their value. In math, you'll often come across expressions that look more complicated than they actually are. Simplifying is the process of rewriting these in a simpler form. To simplify an expression:

• Look for negative exponents or complex fractions that can be transformed.
• Combine like terms, which are terms that have the same variables raised to the same power.

For example, take the expression given here, \( \frac{5}{7 x^{-3/4}} \). By rewriting the negative exponent as a positive one, the expression becomes simpler. Remember, the goal is to keep the expression equivalent but make it more manageable for calculations or further algebraic work. Simplified expressions often make solving problems more efficient and reduce errors.

Negative Exponent Rules

Negative exponents can be a bit tricky at first, but they follow a straightforward rule: any number raised to a negative exponent is the same as one over that number raised to the opposite positive exponent. This is expressed as:\( x^{-a} = \frac{1}{x^a} \)So, if you have \( x^{-3/4} \), you turn it into \( \frac{1}{x^{3/4}} \). This is what we did in our example to simplify the expression.Using this rule often helps in simplifying expressions or equations by converting negative exponents into positive ones. Here are a few key points to remember:

• Negative exponents indicate reciprocation; they "flip" the base number or variable.
• Avoid leaving negative exponents in your final answer whenever possible; it's a common practice in algebra to express using positive exponents.

Fractional Exponents

Fractional exponents represent roots and powers at the same time. A fractional exponent like \( x^{3/4} \) is interpreted as the fourth root of \( x \) cubed. Mathematically, it's equivalent to:\( x^{3/4} = \sqrt[4]{x^3} \)Fractional exponents follow the same rules as regular exponents. You can multiply them, add them, or apply them as needed. In our expression, \( x^{3/4} \) was originally a negative fractional exponent, \( x^{-3/4} \), which we converted to a positive exponent.Important points about fractional exponents:

• The numerator represents the power, and the denominator represents the root.
• Using fractional exponents allows you to combine roots and powers into a single expression, simplifying calculations.