Problem 63 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 6 a^{-4}\left(2 a^{-6}\right) $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$6 a^{-4}\left(2 a^{-6}\right)$$ using only positive exponents. Answer: $$\frac{12}{a^{10}}$$

Step 1: Rewrite negative exponents as positive exponents using the rule $$a^{-n} = \frac{1}{a^n}$$

We rewrite the given expression with positive exponents: $$ 6 a^{-4}\left(2 a^{-6}\right) = 6\cdot\frac{1}{a^4}\cdot2\cdot\frac{1}{a^6} $$

Step 2: Simplify the given expression using the arithmetic operations and exponent rules

We simplify the given expression: $$ 6\cdot\frac{1}{a^4}\cdot2\cdot\frac{1}{a^6} = \frac{6}{a^4}\cdot\frac{2}{a^6} = \frac{12}{a^{4+6}} $$ by combining the fraction and applying the rule $$a^m a^n = a^{m+n}$$.

Step 3: Write the final solution using positive exponents

Our final expression using only positive exponents is: $$ \frac{12}{a^{10}} $$

Key Concepts

Negative Exponents

In mathematics, exponents are used to represent how many times a number, called the base, is multiplied by itself. When this exponent is negative, it might look a bit confusing at first. However, there's a straightforward rule for dealing with negative exponents: any expression raised to a negative exponent means you take the reciprocal of the base and then apply the positive exponent. For example, if you have a base \(a\) raised to the power of -3, this is the same as \(\frac{1}{a^3}\).

When simplifying algebraic expressions, converting negative exponents to positive ones helps in making calculations easier and expressions clear. Just remember the basic transformation:

• \(a^{-n} = \frac{1}{a^n}\)

This rule is pivotal when simplifying or rewriting expressions where negative exponents initially appear.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators such as +, -, *, and /. These expressions are used frequently in mathematics to represent real-world problems and abstract scenarios alike. Variables like \(a\), \(b\), and \(x\) often stand in for unknown or changeable values within these expressions.

The expression \(6 a^{-4}(2 a^{-6})\) includes constants (numbers) and variables raised to exponents. In this case, the variable \(a\) appears with negative exponents, which hint that it will be transformed into a fraction when simplified using the negative exponent rules.

Understanding the structure of algebraic expressions helps in systematically breaking down complex problems into simple, manageable steps so that you can apply mathematical operations and rules effectively.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form, where no further operations can be carried out to change their appearance without altering their value. The step-by-step solution of the example given illustrates how this process works.

In the exercise, the expression \(6 a^{-4}(2 a^{-6})\) is simplified by first rewriting the negative exponents as positive exponents, using the fact that \(a^{-n} = \frac{1}{a^n}\). This converts the expression into \(6 \cdot \frac{1}{a^4} \cdot 2 \cdot \frac{1}{a^6}\).

Next, apply the arithmetic rules: combine the coefficients (numbers) and the like bases by adding their exponents, leading to \( \frac{12}{a^{4+6}}\), which simplifies to \(\frac{12}{a^{10}}\). This is the simplest form of the original expression using only positive exponents.

• Change negative exponents to positive by using reciprocals.
• Combine like terms using basic arithmetic rules.
• Ensure the expression is as simple as possible by reducing where applicable.

These steps are essential in reaching the final simplified form efficiently.