Problem 101 - Intermediate Algebra

Question

Complete the rules for exponents. Assume that there are no divisions by 0. a. \(x^{m} x^{n}=\) b. \(\left(x^{m}\right)^{m}=\) c. \((x y)^{n}=\) d. \(\left(\frac{x}{y}\right)^{n}=\) e. \(x^{0}=\) f. \(x^{-n}=\)

Step-by-Step Solution

Answer

a. \(x^{m+n}\), b. \(x^{m \cdot n}\), c. \(x^{n}y^{n}\), d. \(\frac{x^{n}}{y^{n}}\), e. 1, f. \(\frac{1}{x^{n}}\).

Step 1: Multiplying Powers with the Same Base

When multiplying exponential expressions that have the same base, you add the exponents together. Therefore, for the expression \(x^{m} \cdot x^{n}\), the rule is \(x^{m+n}\).

Step 2: Power of a Power Rule

When raising a power to another power, you multiply the exponents. For the expression \((x^{m})^{n}\), the rule is \(x^{m \cdot n}\).

Step 3: Power of a Product Rule

For a product raised to an exponent, raise each factor in the product to the exponent separately. Thus, \((xy)^{n}\) becomes \(x^{n}y^{n}\).

Step 4: Power of a Quotient Rule

When raising a quotient to an exponent, apply the exponent to both the numerator and denominator separately. Therefore, the expression \(\left(\frac{x}{y}\right)^{n}\) simplifies to \(\frac{x^{n}}{y^{n}}\).

Step 5: Zero Exponent Rule

Any non-zero base raised to the power of zero is equal to 1. That means for the expression \(x^{0}\), the result is 1.

Step 6: Negative Exponent Rule

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. Applying this to \(x^{-n}\), you get \(\frac{1}{x^{n}}\).

Key Concepts

Power of a Power

The "Power of a Power" rule can initially seem tricky, but it's quite straightforward once you understand the basics. When you encounter an expression like \((x^m)^n\), you're dealing with raising a power to another power.

This might make you wonder: what happens to the exponents? Here's the rule in simple terms: you **multiply the exponents**. Therefore, the expression becomes \(x^{m \cdot n}\).

• The base, \(x\), remains the same throughout.
• The operation changes what you do with the exponents, not the base.
• Multiplication of the exponents is the key step here.

For example, if you have an expression like \((x^2)^3\), you multiply the exponents 2 and 3 to get \(x^{6}\). This step simplifies the problem to just evaluating a single power of the base \(x\). Keep this rule in mind as an efficient way to handle multiple levels of exponents.

Zero Exponent Rule

The Zero Exponent Rule is one of the foundational rules for working with exponents and is incredibly useful in simplifying expressions. The rule states: any non-zero base raised to the power of zero is equal to 1. In mathematical terms, \(x^0 = 1\) when \(x\) is not zero.

You might wonder why this is the case. It helps to think about dividing powers to understand why the zero exponent gives us 1:

• Consider \(x^m/x^m\). Following the division rule for exponents, this is \(x^{m-m} = x^0\).
• Since anything divided by itself (other than zero) is 1, \(x^m/x^m = 1\).

Therefore, \(x^0 = 1\).This concept can simplify calculations significantly, especially when dealing with complex expressions that include zero exponents. It's a nifty rule that helps reduce terms and tidy up calculations quickly.

Negative Exponent Rule

The Negative Exponent Rule often causes confusion at first, but it's simple once you grasp the main idea: a negative exponent tells you to take the reciprocal of the base. In specific terms, \(x^{-n} = \frac{1}{x^n}\).

Here's how to think about it:

• The negative sign in the exponent indicates inversion, or flipping, of the base.
• You compute the positive power as usual but in the denominator of a fraction.
• This operation does not change the base itself, just the position — from numerator to denominator.

As an example, consider \(x^{-3}\). Using the rule, this becomes \(\frac{1}{x^3}\). Understanding this will allow you to convert negative exponents into more familiar fractional terms, which are often easier to handle in mathematical expressions.