Problem 87 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{3^{-1} b^{5}(b+7)^{-4}}{9^{-1} a^{-4}(a+7)^{2}} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\frac{\frac{1}{3} b^{5} (b^{-4})}{\frac{1}{9} a^{-4} (a^2)}\) using only positive exponents. Answer: \(\frac{9a^4b^{5}}{3(a+7)^2(b+7)^4}\)

Step 1: Rewrite negative exponents using positive exponents

Applying property 1 to all the negative exponents in the expression, we have: $$ \frac{\frac{1}{3} b^{5} (\frac{1}{(b+7)^4})}{\frac{1}{9} \frac{1}{a^4} (a+7)^{2}} $$

Step 2: Simplify the fractions

Simplify the fractions by multiplying the numerators and denominators separately: $$ \frac{b^{5}}{3(b+7)^4} \times \frac{9a^4}{(a+7)^2} $$

Step 3: Simplify the expression further

Notice that the fractions can be further simplified by canceling out the common terms from the numerator and denominator, like this: $$ \frac{9a^4b^{5}}{3(a+7)^2(b+7)^4} $$

Step 4: Rewrite using positive exponents

Now, we have the expression with only positive exponents, as required: $$ \frac{9a^4b^{5}}{3(a+7)^2(b+7)^4} $$ So, the final expression with all positive exponents is: $$\boxed{ \frac{9a^4b^{5}}{3(a+7)^2(b+7)^4} }.$$

Key Concepts

Negative Exponents

Understanding negative exponents is crucial for simplifying algebraic expressions. A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. For example, when you have a term like \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \).

In the given exercise, negative exponents are present, and the key to rewriting the expression with only positive exponents is to apply this rule. This means that every term with a negative exponent in the numerator moves to the denominator with a positive exponent, and every negative exponent in the denominator moves to the numerator as a positive exponent. This essential step ensures that all exponents in the final expression are positive.

Simplifying Algebraic Expressions

Simplification of algebraic expressions involves several steps that often include the application of exponent rules and combining like terms. Simplifying makes an expression easier to work with and understand.

In our exercise, simplification begins by applying the rule of negative exponents to achieve only positive exponents. Once this is done, we move on to combining and reducing terms. In the case of fractions, this involves multiplying the numerators together and the denominators together, and then further reducing by cancelling out any common factors. This step might also require factoring or expanding expressions to reveal terms that can simplify further.

Properties of Exponents

There are several properties of exponents that are leveraged when simplifying expressions. Common properties include the product rule (\(a^m \cdot a^n = a^{m+n}\)), the quotient rule (\(\frac{a^m}{a^n} = a^{m-n}\)), and the power of a power rule (\((a^m)^n = a^{mn}\)).

In the context of our exercise, these properties are used to rewrite expressions with negative exponents as positive, as well as to simplify and combine terms in both the numerator and the denominator. For example, when we encounter a term in the denominator with a negative exponent, we apply the quotient rule in reverse to move this term to the numerator with a positive exponent. Such properties are the building blocks in the simplification process.