Problem 54 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ \left(\frac{x^{-6}}{y^{-2}}\right)^{-5} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: $$ \left(\frac{x^{-6}}{y^{-2}}\right)^{-5} $$ Answer: The expression rewritten with only positive exponents is: $$ \frac{x^{30}}{y^{10}} $$

Step 1: Distribute power -5 to x^(-6) and y^(-2)

To distribute the power -5, we apply the rule $$ (a^{m})^{n} = a^{mn}$$ to both x^(-6) and y^(-2). $$ \left(\frac{x^{-6}}{y^{-2}}\right)^{-5} = \frac{(x^{-6})^{-5}}{(y^{-2})^{-5}} $$

Step 2: Simplify exponents

Now, multiply the exponents using the rule given above: $$ \frac{(x^{-6})^{-5}}{(y^{-2})^{-5}} = \frac{x^{(-6)(-5)}}{y^{(-2)(-5)}} $$ This simplifies to: $$ \frac{x^{30}}{y^{10}} $$

Step 3: Rewrite exponents using only positive exponents

Since all exponents are already positive, no further simplification is needed. So, the final expression is: $$ \frac{x^{30}}{y^{10}} $$

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with algebraic expressions involving powers. Exponents, or powers, indicate how many times a number or variable is multiplied by itself. For example, the expression \( a^3 \) signifies that \( a \) is multiplied three times: \( a \times a \times a \).

When you encounter a power raised to another power, such as in \( (a^m)^n \), the exponent rules tell you to multiply the exponents. This is known as the power of a power rule, and it is applied by multiplying the exponents, resulting in \( a^{mn} \). Another essential rule is when you multiply two powers with the same base; you add the exponents, leading to \( a^{m+n} \). Conversely, if you divide powers with the same base, you subtract the exponents, forming \( a^{m-n} \).

When simplifying expressions, it's vital to use these rules to rewrite the expression in a form that typically involves positive exponents, which are easier to interpret and work with in subsequent calculations.

Simplifying Exponents

Simplifying exponents is a process that involves applying the exponent rules to make an expression easier to understand and evaluate. For the expression \( \left(\frac{x^{-6}}{y^{-2}}\right)^{-5} \), simplifying the exponents efficiently is pivotal.

In the given expression, the first step is to apply the power to a power rule. This rule states that when you raise an exponent to another exponent, you multiply the two exponents together. Following this rule leads to \( x^{(-6)(-5)} \) and \( y^{(-2)(-5)} \), which simplifies to \( x^{30} \) and \( y^{10} \) respectively. This process transforms a potentially confusing expression with negative exponents into a clear and simplified form with positive exponents. Simplification not only makes the expression neater but also prepares it for further algebraic manipulation or evaluation if necessary.

Algebraic Expressions

Algebraic expressions are combinations of letters (variables), numbers, and mathematical operators that represent a particular value or set of values. When we work with algebraic expressions such as \( \frac{x^{30}}{y^{10}} \), understanding how to handle the variables and exponents is key.

An algebraic expression can become complex when it includes multiple terms with variable bases and negative or fractional exponents. The goal of simplifying such expressions is to rewrite them in the simplest form possible, often incorporating positive exponents for clarity. In the context of exponentiation, positive exponents make an expression straightforward because they directly exemplify multiplication. This contrasts with negative exponents, which imply division and can sometimes confuse the interpretation of the expression. Through the use of exponent rules, algebraic expressions can be simplified effectively, enabling us to work with them in further mathematical processes with greater ease.