Problem 388 Simplify. $$\left(\frac{r^{2}}... [FREE SOLUTION]

Chapter 6: Problem 388

Simplify. $$\left(\frac{r^{2}}{r^{6}}\right)^{3}$$

Short Answer

Expert verified

$ \left(\frac{r^{2}}{r^{6}}\right)^{3} = \frac{1}{r^{12}} $

Step by step solution

Simplify the fraction inside the parentheses

First, simplify the fraction $\frac{r^{2}}{r^{6}}$. When dividing like bases, subtract the exponents: $\frac{r^{2}}{r^{6}} = r^{2-6} = r^{-4}$.

Apply the exponent outside the parentheses

Now that the expression inside the parentheses is simplified, apply the exponent 3 outside the parentheses: $ (r^{-4})^{3} $. When raising a power to another power, multiply the exponents: $ r^{-4 \times 3} = r^{-12} $.

Convert to positive exponent

Finally, convert $ r^{-12} $ with a negative exponent to a fraction using positive exponents: $ r^{-12} = \frac{1}{r^{12}} $.

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Key Concepts

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

Simplifying Exponents

Before diving deeper into complex operations, understanding how to simplify exponents is crucial. Simplifying exponents involves reducing expressions to their simplest form. The basic rule for dividing exponents with the same base is: when dividing, subtract the exponents. For example, in the exercise $\frac{r^2}{r^6}$, you subtract the exponent in the denominator from the exponent in the numerator: $\frac{r^2}{r^6} = r^{2-6} = r^{-4}$. Simplifying exponents helps in making advanced calculations easier and more manageable.

Negative Exponents

Negative exponents might seem tricky, but they follow a straightforward rule. An exponent of $-n$ means you take the reciprocal of the base raised to the positive exponent $n$. In simple terms: $a^{-n} = \frac{1}{a^n}$. For instance, in our solution where $r^{-4}$ is present, the negative exponent can be transformed: $r^{-4} = \frac{1}{r^4}$. This concept ensures we can express all exponents as positive, particularly useful for further simplifications and readability in final answers.

Exponent Multiplication

When dealing with exponent multiplication, it's essential to understand the rule: when raising one exponent to another, you multiply the exponents. For example, given $(r^{-4})^3$, you multiply $-4$ by $3$ to get the new exponent: $-4 \times 3 = -12$. So, $(r^{-4})^3 = r^{-12}$. This rule simplifies expressions involving powers of powers and makes it simpler to manage complex exponential functions in equations and real-world applications. Recognizing this pattern will help streamline many exponent-related problems.

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91影视

Simplify. $$\left(\frac{r^{2}}{r^{6}}\right)^{3}$$

Short Answer

Step by step solution

Simplify the fraction inside the parentheses

Apply the exponent outside the parentheses

Convert to positive exponent

Key Concepts

Simplifying Exponents

Negative Exponents

Exponent Multiplication

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