Problem 57 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ \frac{y^{4}}{x^{-10}} $$

Step-by-Step Solution

Answer

The expression \(\frac{y^{4}}{x^{-10}}\) with positive exponents is \(y^{4}*x^{10}\).

Step 1: Convert Negative Exponent to Positive

The given expression is \(\frac{y^{4}}{x^{-10}}\). The first step involves changing the negative exponent to a positive exponent. According to the rule, x^-m = 1/x^m. Therefore, x^-10 becomes 1/x^10.

Step 2: Simplify the Expression

Next, replace x^-10 in the original expression to get \(y^{4}*x^{10}\). The denominator of the fraction becomes 1 since it is now just a multiplication term.

Step 3: Present the Final Expression

The final expression, after converting the negative exponent to positive, is \(y^{4}*x^{10}\).

Key Concepts

Understanding Negative Exponents

Negative exponents might seem confusing at first glance, but they follow a simple rule. When you see an exponent that is negative, it indicates the reciprocal of the base raised to the opposite positive exponent. For example, if you have a term like \(x^{-n}\), it's equivalent to \(\frac{1}{x^{n}}\). This means that instead of multiplying \(x\) by itself a certain number of times, you're essentially dividing \(1\) by \(x\) to the power of \(n\).
This rule helps in converting expressions with negative exponents into a more standard form, making operations like multiplication and division easier to handle.

• Negative exponents represent reciprocals.
• To simplify, convert to positive exponents using the rule: \(x^{-m} = \frac{1}{x^{m}}\).

How to Rewrite Expressions with Exponents

Rewriting expressions involving exponents is straightforward once you understand the basic rules. When you encounter negative exponents, your goal is to express them with positive exponents for simplification.Consider the expression \(\frac{y^{4}}{x^{-10}}\). Here, the negative exponent \(x^{-10}\) can be converted by taking its reciprocal, resulting in \(x^{10}\). By doing this, the expression is rewritten in a standard and simpler form: \(y^{4} \times x^{10}\). This process not only makes the math cleaner but also prepares terms for further simplifications or other algebraic operations.

• Identify any negative exponents in the expression.
• Apply the rule to convert them to positive exponents.
• Rewrite the expression for clarity.

Simplifying Algebraic Expressions

Algebra simplification is a key skill in solving equations and understanding relationships between variables. It involves reducing expressions to their simplest form while maintaining equality. Once an expression is rewritten with positive exponents, as we saw in \(y^{4} \times x^{10}\), simplifying involves combining like terms if possible and organizing the terms for clarity.In practice, simplification steps generally include:

• Converting negative exponents to positive exponents.
• Combining like terms.
• Using arithmetic operations to consolidate terms.

It's important to work systematically to ensure no mistakes are made and that the expression remains valid. This process is crucial in both solving problems and when preparing expressions for comparison or further calculations in algebra.