Problem 36 - Precalculus Mathematics for Calculus

Question

Simplify each expression. (a) \(x^{-5} x^{3}\) (b) \(w^{-2} w^{-4} w^{6}\) (c) \(z^{5} z^{-3} z^{-1}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{x^2}\), (b) 1, (c) \(z\)

Step 1: Understand the Properties of Exponents

To simplify expressions with exponents, recall the properties of exponents. Specifically, when multiplying like bases, add the exponents: \[ a^m \cdot a^n = a^{m+n}. \] Additionally, for negative exponents, \[ a^{-n} = \frac{1}{a^n}. \] We'll use these rules to simplify each part of the exercise.

Step 2: Simplify (a) by Adding Exponents

For the expression \(x^{-5} x^{3}\), apply the product rule of exponents:\[ x^{-5 + 3} = x^{-2}. \] Since we have a negative exponent, express it as a fraction:\[ x^{-2} = \frac{1}{x^2}. \]

Step 3: Simplify (b) by Adding Exponents

For the expression \(w^{-2} w^{-4} w^{6}\), first add the exponents:\[ w^{-2 + (-4) + 6} = w^{0}. \] Any number raised to the power of zero is 1, so:\[ w^{0} = 1. \]

Step 4: Simplify (c) by Adding Exponents

For the expression \(z^{5} z^{-3} z^{-1}\), use the product rule to add exponents:\[ z^{5 + (-3) + (-1)} = z^{1}. \] Therefore, the expression simplifies to:\[ z^{1} = z. \]

Key Concepts

Negative Exponents

Negative exponents can sometimes seem tricky, but they're actually quite straightforward when you remember one key idea: they indicate division instead of multiplication. Think of a negative exponent as telling you to "flip" the base to the other side of the fraction line and then take the positive exponent. For example, if you see \(a^{-n}\), this means \(\frac{1}{a^n}\). It's like saying "take the reciprocal of \(a\) raised to the \(n\)th power."
In our example from the exercise, we simplified \(x^{-2}\) to \(\frac{1}{x^2}\). This step shows how applying the idea of negative exponents lets us express the power as a fraction, making the expression cleaner and often easier to work with in equations.
Understanding this trick helps you switch from thinking about multiplication to division, which is all negative exponents really are. Once you've got this down, tackling problems with negative exponents becomes a breeze!

Multiplying Exponents

When it comes to multiplying exponents, a simple rule can save you lots of time: if the bases are the same, just add the exponents. This is known as the product of powers property. When you see \(a^m \cdot a^n\), you can simplify it to \(a^{m+n}\).
This rule works because multiplying similar bases is like multiplying the number that many times. For instance, in our exercise, handling \(x^{-5} x^{3}\) meant adding the exponents: \(-5 + 3 = -2\). This led us to \(x^{-2}\). The same method applied to expressions like \(w^{-2} w^{-4} w^{6}\), resulting in a sum of the exponents to ultimately simplify to a nice round \(w^0 = 1\).
By understanding and applying this simple addition rule, you can easily simplify complex-looking products of exponents, turning them into much more manageable forms.

Exponent Rules

Exponent rules are the backbone of simplifying expressions with powers. They aren't just random guidelines; they are based on fundamental math principles. Let's break down the essential rules:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\). This tells you to add exponents when multiplying like bases.
• Power Rule: \((a^m)^n = a^{m \cdot n}\). This is how you handle exponents raised to another power.
• Zero Exponent Rule: Any base raised to the zero power is always 1: \(a^0 = 1\), provided \(a \/= 0\).
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\). Subtract the exponents when dividing like bases.
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\). This shows how to convert a negative exponent into a fraction.

In our exercise, these rules were used to tackle different expressions efficiently. For instance, converting negative exponents or adding up exponents when multiplying similar bases helped simplify the problems. Mastering these fundamental rules equips you to face any exponent-related problem with confidence!