Problem 42 Divide as indicated. Write your ... [FREE SOLUTION]

91影视

Prealgebra

Charles P. McKeague

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 9: Problem 42

Divide as indicated. Write your answer using only positive exponents. $$x^{6} \div x^{-6}$$

Short Answer

Expert verified

The answer is $ x^{12} $.

Step by step solution

Identify the Rule

When dividing expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator. This is known as the Quotient Rule for exponents: $ a^m \div a^n = a^{m-n} $.

Apply the Quotient Rule

Apply the Quotient Rule to the expression $ x^6 \div x^{-6} $. Subtract the exponent in the denominator from the exponent in the numerator: $ 6 - (-6) = 6 + 6 = 12 $.

Write the Final Expression

The simplified form using positive exponents is $ x^{12} $. Since we subtracted a negative, it resulted in adding the exponents.

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Key Concepts

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

Positive Exponents

Exponents are a shorthand way to express how many times a number, known as the base, is multiplied by itself. Positive exponents are straightforward and often the easiest to understand. When an exponent is positive, it simply means:

The base number is multiplied by itself as many times as indicated by the exponent.
Positive exponents do not alter the base number's sign.

For example, with an expression like $x^2$, the base $x$ is multiplied by itself, resulting in $x \times x$. Transitioning from negative to positive exponents might seem tricky at first, but remember: negative exponents indicate division, and converting them to positive simplifies expressions and makes calculations easier. By focusing solely on positive exponents, you ensure clarity in your expressions.

Simplifying Expressions

Simplifying expressions is about making them easier to work with, while maintaining their original value. When dealing with exponents, simplification often involves using rules that apply to the indices. These include:

Applying the power of a product rule.
Using the quotient rule when dividing powers.
Converting negative exponents to positive by changing their position in a fraction.

For instance, in the expression $x^6 \div x^{-6}$, using the quotient rule allows for simplification by subtracting exponents. This results in $x^{12}$. Simplification can transform complex equations into neat, straightforward ones, enabling more accessible computation and comprehension.

Dividing Powers with the Same Base

In mathematics, dividing powers with the same base is efficiently handled by the Quotient Rule for exponents. This rule states: $a^m \div a^n = a^{m-n}$. Here, $a$ is the base, and $m$ and $n$ are the exponents. The process involves:

Recognizing that you're dealing with the same base for both the numerator and denominator.
Subtracting the exponent of the denominator from the exponent of the numerator.

For example, in the expression $x^6 \div x^{-6}$, the bases are both $x$. Subtract the denominator exponent from the numerator exponent: $6 - (-6) = 12$. Therefore, the simplified expression becomes $x^{12}$, immaculately showing the power of the quotient rule in handling divisions seamlessly.

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91影视

Divide as indicated. Write your answer using only positive exponents. $$x^{6} \div x^{-6}$$

Short Answer

Step by step solution

Identify the Rule

Apply the Quotient Rule

Write the Final Expression

Key Concepts

Positive Exponents

Simplifying Expressions

Dividing Powers with the Same Base

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