Problem 82 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{23}{y^{-1}} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\frac{23}{y^{-1}}\) with only positive exponents. Answer: \(23y\).

Step 1: Use the exponent rule

Since there's a negative exponent in the denominator (\(y^{-1}\)), we move it to the numerator to change it into a positive exponent: \((y^{-1})^{-1} = y^{(-1)(-1)} = y^1 = y\)

Step 2: Rewrite the expression with positive exponents

Now, we substitute our result from step 1 back into the original expression to get the rewritten expression with only positive exponents: $$ \frac{23}{y^{-1}}=\frac{23}{1} \times y=\boxed{23y} $$

Key Concepts

Negative Exponents

When you encounter a negative exponent, it can seem puzzling at first. However, understanding this concept is crucial, as negative exponents are common in algebra and various mathematical fields. A negative exponent, such as \(y^{-1}\), indicates that the base, \(y\), should be moved to the opposite position in a fraction. This means if \(y^{-1}\) is in the denominator, moving it to the numerator makes it \(y^1\). The expression \(y^{-1}\) becomes \(\frac{1}{y}\) when it's in the numerator or simply \(y\) when it's moved from the denominator to the numerator. Remember that moving the base to another part of the fraction changes the sign of the exponent from negative to positive.

Exponent Rules

Exponent rules are essential for simplifying expressions and solving equations. There are several fundamental rules you should know:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\). Combine exponents by adding them when multiplying like bases.


• Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\). Subtract the exponents when dividing like bases.


• Power of a Power Rule: \((a^m)^n = a^{m\cdot n}\). Multiply the exponents together when raising a power to another power.


• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\). This flips the position of the base from top to bottom or vice versa in a fraction.

By using these rules, you can handle complex expressions with ease. Negative exponents, in particular, are often simplified by using the Negative Exponent Rule to convert them into positive ones. This helps in simplifying equations and rewriting them in a more manageable form.

Rewriting Expressions

Rewriting expressions may sometimes be necessary to simplify or clarify them. By using the rules of exponents, particularly with negative exponents, expressions can be transformed into an easily interpretable form. Let's take the expression \(\frac{23}{y^{-1}}\) as an example.Initially, the negative exponent \(y^{-1}\) is in the denominator. Using exponent rules, you move \(y^{-1}\) to the numerator and change it to \(y^1\). The expression now becomes \(23y\). This transformation is crucial for simplification, ensuring only positive exponents are present. Rewriting expressions in this way allows you to clearly identify components of the expression and avoids complications from negative exponents. This technique is fundamental for solving equations systematically, leading to more intuitive understanding and clearer presentations of mathematical work.