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Chapter 9: Problem 34

For Exercises, simplify. $$\frac{m}{m^{7}}$$

Short Answer

Expert verified
The simplified form of the given expression is \(m^{-6}\).

Step by step solution

01

Identify the given expression

The given expression to simplify is \(\frac{m}{m^{7}}\)
02

Apply division rule of exponents

Apply the rule for dividing two variables alike with different exponents. In this case, subtract the exponent in the denominator from the exponent in the numerator. Since the exponent of m in the numerator is not explicitly given, it is understood to be 1. Therefore, it becomes \(m^{1-7}\)
03

Simplify the expression

After subtracting, we have \(m^{-6}\), which is the simplified form

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Division Rule of Exponents
When you encounter expressions where exponents with the same base are divided by each other, the division rule of exponents comes into play. The rule states that to divide two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator. This is written as \( a^m \div a^n = a^{m-n} \).

Let's look at the example \( \frac{m}{m^7} \). Here, \(m\) in the numerator is the same as \(m^1\) since any number or variable to the power of 1 is itself. Applying our rule, we subtract the exponents: \( m^{1-7} = m^{-6} \). It's important to remember that this rule applies only when the bases are identical, ensuring the operation is valid.
Algebraic Fractions
Algebraic fractions are fractions that contain algebraic expressions within the numerator, the denominator, or both. Simplifying these fractions often involves applying the rules of exponents among other algebraic properties. The primary goal is to reduce the fraction to its simplest form, potentially making it easier to use in equations and further calculations.

In our given exercise, \( \frac{m}{m^7} \), we're dealing with an algebraic fraction. To simplify, you look for common factors in the numerator and the denominator. Since the base \(m\) is the same in both, we can simplify by using our exponent rules, which leads to \(m^{-6}\) in this case.
Negative Exponents
Negative exponents can be intimidating, but they represent a very simple concept. The negative sign indicates a reciprocal. For any non-zero number or variable \(a\) and any positive integer \(n\), \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\).

In the expression \(m^{-6}\), the negative exponent tells us that the \(m\) is actually on the wrong side of the fraction for our traditional interpretation. To convert it to a positive exponent, we flip it to the denominator, resulting in \(\frac{1}{m^6}\). This is the positive exponent form of our original expression and depicts the same mathematical relationship.

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Most popular questions from this chapter

The electric charge on an electron is 0.00000000000000000016 coulomb. Write this number in scientific notation.

State whether the coefficient of the \(x\) term of the product is positive, negative, or zero. $$(a x+b)(a x-b), \text { where } a>0 \text { and } b>0$$

State whether the coefficient of the \(x\) term of the product is positive, negative, or zero. $$(a x+b)(a x+b), \text { where } a>0 \text { and } b<0$$

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