Problem 61 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 5 x^{3}\left(2 x^{-7}\right) $$

Step-by-Step Solution

Answer

Question: Rewrite the expression with only positive exponents: $5x^3(2x^{-7})$ Answer: $\frac{10}{x^4}$

Step 1: Identify the given expression

The given expression is: $$ 5 x^{3}\left(2 x^{-7}\right) $$

Step 2: Apply the laws of exponents

According to the law of exponents, when multiplying terms with the same base, we add their exponents: $$ a^m \cdot a^n = a^{m+n} $$ Apply this law to the expression: $$ 5 x^{3}\left(2 x^{-7}\right) = 5 \cdot 2 \cdot x^{3}\cdot x^{-7} $$

Step 3: Simplify the expression

Combine the constants and the variables by adding the exponents of the x terms: $$ 5 \cdot 2 \cdot x^{3}\cdot x^{-7} = 10 x^{3-7} $$

Step 4: Calculate the exponent and rewrite the expression with only positive exponents

Now, subtract the exponents of the x terms: $$ 10 x^{3-7} = 10 x^{-4} $$ To write the expression with a positive exponent, use the property: $$ x^{-n} = \frac{1}{x^n} $$ So, the final expression is: $$ 10 x^{-4} = \frac{10}{x^4} $$

Key Concepts

Laws of Exponents

When dealing with mathematical expressions, the laws of exponents are crucial for simplifying and manipulating expressions that involve powers. One of the foundational laws is when you multiply two exponents with the same base, you add their exponents together. This is shown as
\( a^m \times a^n = a^{m+n} \).
This property simplifies multiplication processes and is essential when working with polynomial expressions or any algebraic expressions involving exponents. Understanding and applying these laws correctly can turn complex problems into more manageable ones.

Multiplying Variables

Working with multiplying variables requires recognizing that variables with the same base can be combined using the laws of exponents. Upon facing an expression like
\( x^a \times y^b \), you would treat each variable separately since they have different bases. However, for expressions with the same base, like
\( x^a \times x^b \), you would add the exponents due to the base being the same. In the provided exercise, we applied this rule to multiply \( x^3 \times x^{-7} \) which effectively simplifies to \( x^{3-7} \), by adding the exponents.

Simplifying Algebraic Expressions

The process of simplifying algebraic expressions often involves reducing the expression to its simplest form by performing operations like addition, subtraction, multiplication, and division. The goal is to make the expression as straightforward as possible, often to facilitate solving equations or inequalities. When simplifying expressions with exponents, we combine like terms and apply the laws of exponents to achieve the most basic form of the expression. In our example, after multiplying variables, we combined constants and simplified the expression to \( 10 x^{-4} \) by subtracting the exponents as per the exponent rule.

Exponent Properties

There are several key exponent properties that are used to handle expressions with exponents during simplification. These include the 'product of powers' rule used when multiplying exponents with the same base, the 'power of a power' rule when an exponent expression is raised to another power, and the 'negative exponent' rule which states that any base raised to a negative exponent is equivalent to its reciprocal raised to the corresponding positive exponent: \( a^{-n} = \frac{1}{a^n}\). This last property was used to transform the expression \( 10 x^{-4}\) into a form with only positive exponents as \( \frac{10}{x^4}\). Understanding and correctly applying these properties is essential for working with exponentials.