Problem 80 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{4}{x^{-6}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{4}{x^{-6}}\) using only positive exponents. Answer: \(4x^{6}\)

Step 1: Rewrite the expression using the exponent rule

To rewrite the expression using the exponent rule, we can apply the rule \(a^{-n} = \frac{1}{a^n}\) to simplify \(x^{-6}\). In this case, \(a=x\) and \(n=6\): $$ \frac{4}{x^{-6}} = \frac{4}{\frac{1}{x^{6}}} $$

Step 2: Multiply by the reciprocal

To simplify the expression further, we multiply the numerator and denominator by the reciprocal of the denominator: $$ \frac{4}{\frac{1}{x^{6}}} = \frac{4}{1} \cdot \frac{x^{6}}{1} $$

Step 3: Simplify the expression

Now, we can simplify the expression by multiplying the numbers and variables: $$ \frac{4}{1} \cdot \frac{x^{6}}{1} = 4x^{6} $$ So the simplified expression using only positive exponents is \(4x^{6}\).

Key Concepts

Exponent Rule

The exponent rule is a fundamental principle in algebra that helps us manipulate and simplify expressions involving exponents. An exponent refers to the number of times a number, known as the base, is multiplied by itself. Sometimes, exponents can be negative, like in the problem with \( x^{-6} \).

The rule \( a^{-n} = \frac{1}{a^n} \) tells us that a negative exponent means we take the reciprocal of the base raised to the positive exponent. In simpler terms, this rule allows us to convert negative exponents into positive ones by flipping the base. This is especially helpful in ensuring that all exponents in the final expression are positive, making it much simpler to understand and work with.

So, whenever you encounter a negative exponent, remember this handy rule and transform it into a more manageable form. This conversion makes subsequent algebraic manipulations easier and keeps your math expressions neat.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest and most concise form. This process helps to make the expressions easier to understand and to work with, especially when dealing with complex equations.

In our example, we started with the expression \( \frac{4}{x^{-6}} \). Using the exponent rule, we converted \( x^{-6} \) into \( \frac{1}{x^6} \). Next, we must simplify this complex fraction.

Multiplying by the reciprocal of \( \frac{1}{x^6} \) is the key step here. By multiplying \( \frac{4}{1} \) by \( \frac{x^6}{1} \), we eliminate any fractions and are left with \( 4x^6 \).

Remember, simplifying isn't just about following steps; it's about recognizing opportunities to make expressions cleaner and easier to interpret. It involves looking for patterns and knowing the algebraic rules to apply at each step.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division) that represent a value or set of values. Understanding how to manipulate and simplify these expressions is vital in algebra.

Each part of an algebraic expression holds a significant role. In the expression \( 4x^6 \), \( 4 \) is the coefficient, while \( x^6 \) is a term with a variable base \( x \) and an exponent. Our primary goal often is to simplify such expressions into their most reduced forms, using rules like the exponent rule.

Working with algebraic expressions also means being comfortable with operations on variables just as with real numbers. This ensures you can handle similar problems, combining terms and breaking them down accurately. The practice builds a solid foundation for tackling more advanced algebraic concepts.