Problem 102 - Intermediate Algebra

Question

Complete the rules for exponents. Assume that there are no divisions by 0. a. \(\frac{x^{m}}{x^{n}}=\) b. \(\left(\frac{x}{y}\right)^{-n}=(\quad)\) c. \(\frac{x^{-m}}{y^{-n}}=\) d. \(x^{1}=\)

Step-by-Step Solution

Answer

a. \(x^{m-n}\), b. \((\frac{y}{x})^n\), c. \(\frac{y^n}{x^m}\), d. \(x\).

Step 1: Simplify the Fractional Exponent - Part a

The rule for dividing powers with the same base involves subtracting the exponents. Therefore, \( \frac{x^m}{x^n} = x^{m-n} \). This rule applies when the base \( x \) is the same for both numerator and denominator.

Step 2: Invert the Base - Part b

To solve \( \left(\frac{x}{y}\right)^{-n} \), we use the negative exponent rule, which states that a negative exponent indicates taking the reciprocal of the base. Thus, \( \left(\frac{x}{y}\right)^{-n} = \left(\frac{y}{x}\right)^n \).

Step 3: Convert Negative Exponents - Part c

For \( \frac{x^{-m}}{y^{-n}} \), we use the rule that \( x^{-a} = \frac{1}{x^a} \). Consequently, \( \frac{x^{-m}}{y^{-n}} = \frac{1/x^m}{1/y^n} = \frac{y^n}{x^m} \). This involves flipping the exponents to positive by inverting their bases.

Step 4: Simplify Identity Exponent - Part d

The rule of identity exponent states that any number raised to the power of 1 is the number itself. Thus, \( x^1 = x \).

Key Concepts

Negative Exponents

Negative exponents can be a bit tricky at first, but they have a simple rule. When you see a negative exponent, it means that you need to take the reciprocal of the base and then raise it to the positive of that exponent.
For example, if we have something like \( x^{-n} \), it's the same as saying \( \frac{1}{x^n} \). This rule helps when simplifying expressions with negative exponents, like \( \frac{x^{-m}}{y^{-n}} \).

• First, take the reciprocal of the bases \( x \) and \( y \) individually, which results in \( \frac{1}{x^m} \) and \( \frac{1}{y^n} \).
• Then, rearrange these fractions to \( \frac{y^n}{x^m} \).

This rule makes calculations easier and helps in transforming various mathematical expressions.

Fractional Exponents

Fractional exponents might look intimidating, but they are just another way to express roots and powers in math. When you have a fractional exponent like \( x^{\frac{m}{n}} \), it means you are dealing with both an exponent and a root.
In this case:

• \( x^{\frac{m}{n}} \) is equivalent to \( \sqrt[n]{x^m} \).
• Here, \( n \) is the root, and \( m \) is the exponent of the base \( x \).

Understanding fractional exponents is crucial because it links exponents with roots, making it easier to solve complex equations and provides flexibility in simplifying mathematical expressions.

Identity Exponent

The concept of identity exponent is one of the simplest in algebra. Whenever you have a number raised to the power of 1, like \( x^1 \), the number remains the same.
This is because any number multiplied by 1 is itself. So:

• For \( x^1 \), the result will simply be \( x \).
• This rule is fundamental and holds for any base, whether it's a number or a variable.

Understanding this rule is key to recognizing and simplifying many mathematical equations. It may seem obvious, but it plays a critical role in forming the basis for more complex exponent rules.

Dividing Powers

When dividing powers that have the same base, it's important to remember the rule about subtracting exponents. This rule states that \( \frac{x^m}{x^n} = x^{m-n} \).
This applies when the bases are the same, making it easier to simplify expressions.

• Ensure both terms in the division have the same base \( x \).
• Subtract the exponent of the denominator from the exponent of the numerator.
• The result is a power of \( x \), elevated to the difference between the exponents \( m-n \).

These rules are handy for simplifying algebraic expressions and solving equations that involve powers. Keeping them in mind will make dividing powers straightforward.