Problem 75 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (-9)^{-3}(9)^{3} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: $$(-9)^{-3}(9)^3$$ Answer: The expression can be simplified to $$(-1)^3 = -1$$ using exponent properties.

Step 1: Simplify the expression using exponent properties

First, we need to recall the rule for dealing with negative exponents: $$ a^{-n} = \frac{1}{a^n} $$ Apply this rule to the first term \((-9)^{-3}\) in the given expression: $$ (-9)^{-3} = \frac{1}{(-9)^3} $$ Now our expression is: $$ \frac{1}{(-9)^3}(9)^3 $$

Step 2: Continue simplifying the expression

Note that \((-9)^3\) and \((9)^3\) have the same exponent. We can use the property: $$ \frac{a^n}{b^n} = (\frac{a}{b})^n $$ Applying this property to our expression, we get: $$ \frac{1}{(-9)^3}(9)^3 = (\frac{9}{-9})^3 $$

Step 3: Simplify the fraction

Now, simplify the fraction inside the parentheses: $$ \frac{9}{-9} = -1 $$ So, our expression becomes: $$ (-1)^3 $$

Step 4: Evaluate the final expression

Finally, calculate \((-1)^3\): $$ (-1)^3 = -1 $$ Therefore, the given expression can be simplified to \(-1\), which is a constant with a positive exponent.

Key Concepts

Negative Exponents

Negative exponents can sometimes be confusing, but they are easier than they seem. Essentially, a negative exponent tells us to take the reciprocal of the base. In mathematical terms, this means:

• If you have a number or variable with a negative exponent, like \(a^{-n}\), you rewrite it as \(\frac{1}{a^n}\).
• This property helps to convert negative exponents into positive exponents, simplifying expressions and computations.

For example, in the expression \((-9)^{-3}\), we apply the negative exponent rule and rewrite it as \(\frac{1}{(-9)^3}\). This conversion helps us see the expression in terms of division, which is often easier to understand and work with. Understanding negative exponents is crucial, as it opens the door to a more comprehensive understanding of algebraic simplification.

Positive Exponents

Positive exponents are straightforward. They indicate how many times we multiply the base by itself. For instance, in \(a^3\), the positive exponent 3 tells us to multiply the base \(a\) by itself three times: \(a \times a \times a\).

• To maintain expressions with positive exponents, we sometimes use exponent rules like the multiplication property.
• This property states that \(a^n \times a^m = a^{n+m}\).

In the original exercise, the expression \((9)^3\) uses positive exponents directly and doesn't require conversion like negative exponents do. Embracing positive exponents from the start helps in calculating and simplifying expressions smoothly. It's important to get comfortable with positive exponents as they connect directly to understanding larger operations and simplifying tasks.

Simplifying Expressions

Simplifying expressions is key in mathematics to make equations easier to understand and solve. The goal is to reduce an expression to its simplest form. Here are some general steps and tips you can follow:

• Identify terms with exponents, whether they are positive or negative.
• Apply exponent rules to rewrite or combine terms when possible.
• Combine like terms or simplify fractions to reduce the number of components in the expression.

In the exercise, we simplified \((-9)^{-3}(9)^{3}\) by using exponent properties like the negative exponent rule and fraction simplification. Ultimately, we reduced the expression from complex to the single number \(-1\). By practicing these methods, you'll gain confidence in handling even more complicated expressions and equations. Whether you're simplifying homework problems or tackling advanced math, knowing how to simplify expressions is a fundamental skill.