Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
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![### Rational Exponents
**Problem:**
Rewrite each of the following using rational exponents.
**Expressions:**
1. \( \left( \sqrt[4]{x} \right)^2 \)
2. \( \sqrt[5]{x} \)
3. \( \left( \sqrt{x} \right)^7 \)
**Detailed Explanation:**
1. **Expression: \( \left( \sqrt[4]{x} \right)^2 \)**
To express using rational exponents, note that the \( n \)-th root of \( x \) is written as \( x^{1/n} \). Thus, the fourth root of \( x \) can be written as:
\[
\sqrt[4]{x} = x^{\frac{1}{4}}
\]
Since the expression is raised to the power of 2, we apply the exponent rule \( (a^m)^n = a^{mn} \):
\[
\left( x^{\frac{1}{4}} \right)^2 = x^{\frac{1}{4} \cdot 2} = x^{\frac{2}{4}} = x^{\frac{1}{2}}
\]
2. **Expression: \( \sqrt[5]{x} \)**
Using the same principle as above, the fifth root of \( x \) is written with rational exponents as:
\[
\sqrt[5]{x} = x^{\frac{1}{5}}
\]
3. **Expression: \( \left( \sqrt{x} \right)^7 \)**
The square root of \( x \) can be written as \( x^{1/2} \). Raising this expression to the power of 7 gives:
\[
\left( x^{\frac{1}{2}} \right)^7 = x^{\frac{1}{2} \cdot 7} = x^{\frac{7}{2}}
\]
Thus, the expressions rewritten using rational exponents are:
1. \( \left( \sqrt[4]{x} \right)^2 = x^{\frac{1}{2}} \)
2. \( \sqrt[5]{x} = x^{\frac{1}{5}} \)
3. \( \left( \sqrt{x} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd23b2e37-8ff8-4dcd-9ea0-dfa5d98ae8ea%2F48e5b9dd-bc19-48bf-8793-0f0adb0b7f48%2Fdh4zbgs_processed.png&w=3840&q=75)
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