Problem 59 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (x-5)^{-4} 3 b^{2} c^{4}(x+6)^{8} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \((x-5)^{-4} \cdot 3 \cdot b^2 \cdot c^4 \cdot (x+6)^8\) using only positive exponents. Answer: \(\frac{3 b^2 c^4 (x+6)^8}{(x-5)^4}\)

Step 1: Rewrite the term with a negative exponent as a fraction with a positive exponent

To rewrite the term with a negative exponent, we will use the property \(a^{-n}=\frac{1}{a^n}\). In our case, the term is \((x-5)^{-4}\). Applying the property, we have: \[ (x-5)^{-4} = \frac{1}{(x-5)^4} \]

Step 2: Substitute the rewritten term back into the expression

Now we will substitute the modified term back into the expression and simplify: \[ \frac{1}{(x-5)^4} \cdot 3 \cdot b^2 \cdot c^4 \cdot (x+6)^8 \]

Step 3: Write the final expression

After substitution, we can rewrite the expression as a single fraction that contains only positive exponents: \[ \frac{3 b^2 c^4 (x+6)^8}{(x-5)^4} \] The given expression has been successfully rewritten using only positive exponents.

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they're a versatile tool in mathematics that can be easily understood with the right explanation. A negative exponent indicates that the base of the term should be inverted or flipped. This doesn't mean you make the number negative, only that you take the reciprocal of the base. For example, with the expression \( a^{-n} \), this means you would rewrite it as \( \frac{1}{a^n} \). So, rather than imagining that the base itself is negative, think of the negative sign as indicating a shift in position—from the numerator to the denominator.
Using negative exponents allows mathematicians to handle division and simplification in a more compact form. In our exercise, we observe the term \((x-5)^{-4}\). We transform it using this property to \(\frac{1}{(x-5)^4}\). This conversion is crucial because it harmonizes all the elements of the expression into the same notation, making calculations much simpler.

Exponent Rules

Understanding exponent rules is key for simplifying expressions and solving equations accurately. These rules allow us to work with powers effectively and combine them in various ways. Two essential rules to keep in mind are the Product of Powers and the Power of a Power.
\(a^m \cdot a^n = a^{m+n}\) is the Product of Powers rule, which says if you multiply two powers with the same base, you add the exponents. This makes it much easier to reduce complex expressions by combining them into simpler terms.
For Power of a Power, the rule is \((a^m)^n = a^{m \cdot n}\). This indicates that when raising an exponent to another power, you multiply the exponents together.
These rules become even more powerful when combined with others, such as the Negative Exponent rule, which we've already discussed. Mastery of these rules equips you with the tools to tackle a variety of problems, simplifying expressions with confidence and accuracy.

Fraction with Exponent

When you work with fractions that have exponents, you're fundamentally dealing with the properties of both fractions and exponents together. A fraction with an exponent requires you to apply the exponent to both the numerator and the denominator. For example, \((\frac{a}{b})^n = \frac{a^n}{b^n}\). This property is important because it maintains the relationships between the numerator and the denominator when scaling them by the exponent.
In the given exercise, after converting the negative exponent, we are left with \(\frac{1}{(x-5)^4}\). Then, returning to the broader expression, it's clear how we consolidate all factors into a single fraction by separating constants and powers. By understanding how to manage fractions and exponents simultaneously, you can simplify and manage expressions with greater ease.
Remember, every part of the fraction gets raised to the power, which simplifies calculations and ensures that the expression remains balanced.