Question

Simplify the expression.\(\frac{5 z^{5}}{z^{7}}\)

Short answer

The simplified expression of \(\frac{5 z^{5}}{z^{7}}\) is \(\frac{5}{z^{2}}\).

Step-by-step solution

Understand the problem

The aim is to simplify the expression \(\frac{5 z^{5}}{z^{7}}\). The rule to be used is that when dividing numbers of the same base, you subtract their exponents.

Subtract the exponents

Start by subtracting the exponents of \(z\). Write down the new exponent of \(z\) by subtracting the exponent from the denominator from the exponent in the numerator: \(5 - 7 = -2\). So the simplified expression will be \(5z^{-2}\).

Finalize the expression

The above expression can be left as is, but sometimes it is preferred not to leave negative exponents. Therefore, \(5z^{-2}\) can be rewritten as \(\frac{5}{z^{2}}\).

Key concepts

Exponent Rules

When working with expressions that contain exponents, it's important to remember some basic rules. These rules allow us to simplify expressions involving multiplication and division of terms with the same base.

• Product of Powers: When multiplying terms with the same base, you add the exponents. For example, if you have \(a^m \times a^n\), this equals \(a^{m+n}\).
• Quotient of Powers: When dividing terms with the same base, you subtract the exponent in the denominator from the exponent in the numerator. For instance, \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: When raising a power to another power, you multiply the exponents. For instance, \((a^m)^n = a^{m\times n}\).

In the original exercise, the quotient of powers rule was used to simplify \(\frac{5z^5}{z^7}\) by subtracting the exponents: \(5 - 7 = -2\). Thus, the expression simplifies to \(5z^{-2}\). Remembering these rules is key to mastering exponent-related problems.

Negative Exponents

Negative exponents can sometimes be tricky because they introduce a concept that seems to invert our usual understanding of "powers." Instead of making a number larger, the exponent turns it smaller through a process of reciprocal operations.The rule to understand negative exponents is straightforward:

• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\). This tells us that any base raised to a negative exponent is equal to the reciprocal of the base raised to the positive of that exponent.

To convert \(5z^{-2}\), provided in the solution, into a non-negative exponent form, rewrite it as \(\frac{5}{z^2}\). Negative exponents can therefore be seen as instructions to "flip" the base into a denominator under 1, which helps make expressions easier to handle and understand.

Rational Expressions

Rational expressions are similar to fractions, but instead of numbers in the numerator and denominator, they contain polynomials. Simplifying rational expressions involves applying the same principles used with numerical fractions, but we often need to apply exponent rules as well.

• Simplifying Rational Expressions: When you simplify rational expressions, you look to divide both the numerator and the denominator by any common factors. This step often involves simplifying exponents, just as in our original exercise.
• Minding the Domain: Remember, denominators cannot be zero, since division by zero is undefined. Always check what values might make the denominator zero and consider stating them as restrictions.

For example, the expression \(\frac{5z^5}{z^7}\) is a simple rational expression. Once simplified using exponent rules (resulting in \(\frac{5}{z^2}\)), we observe it in a clearer form without negative exponents, making further operations or interpretations easier. Understanding rational expressions aids in more complex algebra manipulations and problem-solving.