Question
Write each expression with a single rational exponent. Show each step of your process. Which expressions are equivalent? Justify your reasoning.
fourth root of x cubed
one over x to the negative one power
tenth root of the quantity x to the fifth times x to the fourth times x squared end quantity
x to the one third power times x to the one third power times x to the one third power
Answers
Answered by
GPT-4o mini
To express each of the given expressions with a single rational exponent, we will follow these steps:
### 1. Fourth root of \(x^3\)
The fourth root of a number can be expressed as that number raised to the power of \(1/4\). Thus, the fourth root of \(x^3\) can be expressed as:
\[
\sqrt[4]{x^3} = (x^3)^{1/4}
\]
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we get:
\[
(x^3)^{1/4} = x^{3 \cdot \frac{1}{4}} = x^{\frac{3}{4}}
\]
### 2. One over \(x^{-1}\)
This expression can be rewritten using the property of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\):
\[
\frac{1}{x^{-1}} = x^{1}
\]
### 3. Tenth root of the quantity \(x^5 \cdot x^4 \cdot x^2 \)
First, simplify the expression inside the parentheses. We can combine the exponents by adding them (since \(x^m \cdot x^n = x^{m+n}\)):
\[
x^5 \cdot x^4 \cdot x^2 = x^{5 + 4 + 2} = x^{11}
\]
Now, we express the tenth root:
\[
\sqrt[10]{x^{11}} = (x^{11})^{1/10}
\]
Using the power of a power rule again:
\[
(x^{11})^{1/10} = x^{\frac{11}{10}}
\]
### 4. \(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)
Use the property of exponents (as shown previously) to combine the terms:
\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = x^{\frac{1}{3} \cdot 3} = x^{1}
\]
### Summary of Expressions with Single Rational Exponents:
1. Fourth root of \(x^3\) can be rewritten as:
\[x^{\frac{3}{4}}\]
2. One over \(x^{-1}\) can be rewritten as:
\[x^{1}\]
3. Tenth root of \(x^5 \cdot x^4 \cdot x^2\) can be rewritten as:
\[x^{\frac{11}{10}}\]
4. \(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\) can be rewritten as:
\[x^{1}\]
### Which Expressions are Equivalent?
Now we can identify which expressions are equivalent based on their final forms:
- The expression \(x^{1}\) appears from both the second expression (\(1/x^{-1}\)) and the fourth expression (\(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)).
- The expressions \(x^{\frac{3}{4}}\) and \(x^{\frac{11}{10}}\) are different from each other and from \(x^1\).
Thus, the equivalent expressions are:
- **One over \(x^{-1}\)** and **\(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)** because both simplify to \(x\).
### 1. Fourth root of \(x^3\)
The fourth root of a number can be expressed as that number raised to the power of \(1/4\). Thus, the fourth root of \(x^3\) can be expressed as:
\[
\sqrt[4]{x^3} = (x^3)^{1/4}
\]
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we get:
\[
(x^3)^{1/4} = x^{3 \cdot \frac{1}{4}} = x^{\frac{3}{4}}
\]
### 2. One over \(x^{-1}\)
This expression can be rewritten using the property of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\):
\[
\frac{1}{x^{-1}} = x^{1}
\]
### 3. Tenth root of the quantity \(x^5 \cdot x^4 \cdot x^2 \)
First, simplify the expression inside the parentheses. We can combine the exponents by adding them (since \(x^m \cdot x^n = x^{m+n}\)):
\[
x^5 \cdot x^4 \cdot x^2 = x^{5 + 4 + 2} = x^{11}
\]
Now, we express the tenth root:
\[
\sqrt[10]{x^{11}} = (x^{11})^{1/10}
\]
Using the power of a power rule again:
\[
(x^{11})^{1/10} = x^{\frac{11}{10}}
\]
### 4. \(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)
Use the property of exponents (as shown previously) to combine the terms:
\[
x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = x^{\frac{1}{3} \cdot 3} = x^{1}
\]
### Summary of Expressions with Single Rational Exponents:
1. Fourth root of \(x^3\) can be rewritten as:
\[x^{\frac{3}{4}}\]
2. One over \(x^{-1}\) can be rewritten as:
\[x^{1}\]
3. Tenth root of \(x^5 \cdot x^4 \cdot x^2\) can be rewritten as:
\[x^{\frac{11}{10}}\]
4. \(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\) can be rewritten as:
\[x^{1}\]
### Which Expressions are Equivalent?
Now we can identify which expressions are equivalent based on their final forms:
- The expression \(x^{1}\) appears from both the second expression (\(1/x^{-1}\)) and the fourth expression (\(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)).
- The expressions \(x^{\frac{3}{4}}\) and \(x^{\frac{11}{10}}\) are different from each other and from \(x^1\).
Thus, the equivalent expressions are:
- **One over \(x^{-1}\)** and **\(x^{1/3} \cdot x^{1/3} \cdot x^{1/3}\)** because both simplify to \(x\).