Problem 70 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 8(b+2)^{-8}(b+2)^{-4}(b+2)^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression 8(b+2)^{-8}(b+2)^{-4}(b+2)^{3} using only positive exponents and nonzero variables. Answer: The simplified expression is $$\frac{8}{(b+2)^9}$$.

Step 1: Combine the terms with identical bases

As per the product of powers property, we know that $$a^{m} \cdot a^{n} = a^{m+n}$$. In our given expression, we have terms with the base \((b+2)\). So, we will use the product of powers property to combine these terms: $$ 8(b+2)^{-8}(b+2)^{-4}(b+2)^{3} = 8(b+2)^{-8-4+3} $$

Step 2: Simplify the exponents

Now, we will simplify the exponents in the expression: $$ 8(b+2)^{-8-4+3} = 8(b+2)^{-9} $$

Step 3: Write the negative exponent as a positive exponent using the reciprocal rule

According to the reciprocal rule, if $$a^{-n} = \frac{1}{a^n}$$, then we can rewrite the expression with a positive exponent as follows: $$ 8(b+2)^{-9} = 8\frac{1}{(b+2)^9} $$

Step 4: Write the simplified expression using only positive exponents

Finally, we can rewrite the expression with only positive exponents: $$ 8\frac{1}{(b+2)^9} = \frac{8}{(b+2)^9} $$ So the given expression using only positive exponents is: $$ \frac{8}{(b+2)^9} $$

Key Concepts

Product of Powers Property

The product of powers property is a fundamental rule in algebra that helps simplify expressions involving exponents. When multiplying terms that have the same base, we add their exponents. The mathematical representation of this is: \(a^{m} \cdot a^{n} = a^{m+n}\). This property is handy because it allows us to condense multiple exponential expressions into a single term.

In our original exercise, the expression \((b+2)^{-8}(b+2)^{-4}(b+2)^{3}\) showcases a perfect example of applying this property. Since all these terms share the base \( (b+2) \), we can add their exponents together: \(-8, -4,\) and \(3\). As a result, we get \((b+2)^{-8-4+3} = (b+2)^{-9}\).

By using the product of powers property, we have efficiently combined multiple exponential terms into a single term, significantly simplifying the expression.

Negative Exponents

Negative exponents can sometimes seem intimidating, but they are quite simple to understand once you grasp the concept. When you see a negative exponent, it indicates the reciprocal of the base raised to the corresponding positive exponent. In simpler terms, \(a^{-n} = \frac{1}{a^n}\).

Going back to our exercise, after using the product of powers property, we have \((b+2)^{-9}\). The negative exponent here means that instead of multiplying \((b+2)\) nine times, we are essentially considering the reciprocal of \((b+2)\) raised to the ninth power.

Understanding negative exponents is crucial since they often appear in various mathematical contexts. Once we recognize what they represent, we can further simplify expressions by converting them to their reciprocal forms.

Reciprocal Rule

The reciprocal rule is closely tied to the concept of negative exponents. As mentioned earlier, a negative exponent like \(a^{-n}\) is, in fact, the reciprocal of the base raised to a positive exponent: \(\frac{1}{a^n}\). This conversion is often necessary to ensure that all exponents in an expression are positive.

In the exercise, we used this rule to transform \((b+2)^{-9}\) into \(\frac{1}{(b+2)^9}\). This step clarifies the expression further and eliminates any negative exponents, which is often a requirement when simplifying expressions.

The reciprocal rule is an algebraic tool that facilitates the simplification of expressions and ensures clarity, especially in equations where interpretations might depend on the positivity of exponents.

Simplifying Expressions

Simplifying expressions is the process of making mathematical statements as efficient as possible. This involves reducing the number of terms, combining like terms, and eliminating unnecessary complexity. One aim of simplification is to produce an expression that is easier to understand and use.

In our problem, simplifying involved several steps. We first used the product of powers property to combine the terms with the same base. Next, we addressed the negative exponents by employing the reciprocal rule. Finally, we rewrote the expression only with positive exponents, resulting in \(\frac{8}{(b+2)^9}\).

By simplifying expressions, we make them more streamlined and manageable. This not only helps in solving equations more easily but also aids in communicating mathematical ideas more effectively. Simplification is a fundamental skill in algebra and is invaluable for both students and professionals.