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Chapter 7: Problem 90

Simplify. Assume that all variables are nonnegative. $$(\sqrt{r^{3}} t)^{7}$$

Short Answer

Expert verified
The simplified form is \( r^{21/2} t^{7} \).

Step by step solution

01

- Simplify the Expression Inside the Parentheses

First, simplify the expression inside the parentheses. The expression is \(\sqrt{r^{3}} t\). Since \(\sqrt{r^{3}}\) can be written as \(r^{3/2}\), the expression becomes \(r^{3/2} t\).
02

- Apply the Exponentiation Rule

Next, apply the exponentiation rule \( (ab)^{c} = a^{c} b^{c} \). Using this rule, \( (r^{3/2} t)^{7} = (r^{3/2})^{7} (t)^{7} \).
03

- Simplify Powers of a Power

Now, simplify each term by applying the rule \( (a^{m})^{n} = a^{m \cdot n} \). So, \( (r^{3/2})^{7} = r^{(3/2) \cdot 7} = r^{21/2} \)Similarly, \( (t)^{7} = t^{7} \).
04

- Combine the Simplified Terms

Combine the simplified terms to get the final expression. Therefore, \( (\sqrt{r^{3}} t)^{7} = r^{21/2} t^{7} \).

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Key Concepts

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

Exponent Rules
Exponent rules are essential for simplifying expressions involving powers. One key rule is the exponentiation rule, which states that when you raise a product to a power, you can distribute the exponent to each factor. This is written as \((ab)^{c} = a^{c} b^{c}\).
Another important rule is the power of a power rule. When you have an exponent raised to another exponent, you multiply the exponents. For instance, \((a^{m})^{n} = a^{m \times n}\).
Understanding these rules is crucial, as they allow you to simplify complex expressions and perform calculations more easily. These rules also help in converting radical forms to exponential forms for easier manipulation.
Fractional Exponents
Fractional exponents are an alternative way to express roots. For example, the square root of a number can be written using a fractional exponent as follows: \(\frac{1}{2}\). So, \(\sqrt{r} = r^{1/2}\).
Likewise, the cube root of a number can be written as \(r^{1/3}\). In our exercise, we encounter \(r^{3/2}\), which represents \(\frac{3}{2}\) power of \(r\).
This is very useful for simplifying expressions, as it allows us to use exponent rules more effectively. By rewriting roots as fractional exponents, we can simplify the entire expression more easily using exponentiation rules.
Simplification
Simplification is the process of reducing an expression to its simplest form. In our exercise, we start by converting the given roots to fractional exponents. For instance, \(\sqrt{r^{3}}\) converts to \(r^{3/2}\). This makes it easier to handle the exponentiation later.
Next, we apply the exponentiation rule to distribute the outer exponent across each term in the parentheses. We then use the power of a power rule to simplify each term further.
Finally, we combine the simplified terms to get our final expression, which is easier to understand and work with. Simplification is crucial for solving algebraic expressions efficiently and accurately, ensuring we get the correct result.

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Most popular questions from this chapter

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