Problem 96 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{4 x^{3}}{y^{7}} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression with only positive exponents: $$\frac{4x^3}{y^7}$$ Answer: The expression with only positive exponents is: $$\frac{4x^3}{y^7}$$

Step 1: Identify negative exponents

Check the expression for any negative exponents: $$ \frac{4x^3}{y^7} $$ No negative exponents are present in the expression. Step 2: Rewrite the expression

Step 2: Rewrite the expression

Since there are no negative exponents in the expression, we can present it as a final answer. The expression with only positive exponents is: $$ \frac{4x^3}{y^7} $$

Key Concepts

Exponent Rules

Understanding exponent rules is essential when dealing with algebraic expressions. Exponents, also known as powers, represent how many times a number is multiplied by itself. For example, the expression \( 2^3 \) means \( 2 \) is multiplied by itself three times: \( 2 \times 2 \times 2 = 8 \).

There are several basic exponent rules that are widely used:

• Product Rule: \( a^m \times a^n = a^{m+n} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \), where \( a eq 0 \)
• Power Rule: \( (a^m)^n = a^{m \times n} \)
• Zero Exponent: \( a^0 = 1 \), where \( a eq 0 \)
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \), where \( a eq 0 \)

These rules are the cornerstone for simplifying algebraic expressions and solving complex problems involving powers. It is important to note that these rules apply to expressions with like bases.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They are the foundation of algebra and are used to represent real-world problems in mathematical terms. An algebraic expression doesn't have an equality sign, unlike an equation.

For instance, \( 4x^3 \) and \( y^7 \) are both algebraic expressions. When combined as \( \frac{4x^3}{y^7} \), we get a more complex expression involving both multiplication and division. Handling algebraic expressions requires an understanding of exponent rules, distributive properties, combining like terms, and factoring. In the exercise given, \( \frac{4x^3}{y^7} \) is already in its simplest form, assuming all variables are nonzero.

Negative Exponents

Negative exponents are often encountered in algebra and they follow a specific rule that converts them into positive exponents. The negative exponent rule states that for any nonzero number \( a \) and a positive integer \( n \), the expression with a negative exponent can be rewritten as \( a^{-n} = \frac{1}{a^n} \).

This implies that any term with a negative exponent in the numerator moves to the denominator with a positive exponent, and vice versa. For example, \( 2^{-3} \) is the same as \( \frac{1}{2^3} \) or \( \frac{1}{8} \). Similarly, if the expression was \( \frac{1}{y^{-7}} \), it would simplify to \( y^7 \) by applying the negative exponent rule. The exercise provided does not have any negative exponents, so no further conversion is necessary. Nevertheless, understanding how to manipulate negative exponents is crucial when simplifying algebraic expressions.