Certainly! Let's simplify the expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] and rewrite it in its simplest rational exponent form by following a detailed, step-by-step process.
### Step 1: Rewrite the radicals using rational exponents
First, we need to express the given radicals as exponents:
- The square root of [tex]\(x\)[/tex], denoted [tex]\(\sqrt{x}\)[/tex], can be rewritten as [tex]\(x^{\frac{1}{2}}\)[/tex].
- The fourth root of [tex]\(x\)[/tex], denoted [tex]\(\sqrt[4]{x}\)[/tex], can be rewritten as [tex]\(x^{\frac{1}{4}}\)[/tex].
So, the expression becomes:
[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{4}}
\][/tex]
### Step 2: Use the property of exponents to combine the terms
We use the property of exponents that states:
[tex]\[
a^m \cdot a^n = a^{m+n}
\][/tex]
Applying this property to our expression, we get:
[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} = x^{\left(\frac{1}{2} + \frac{1}{4}\right)}
\][/tex]
### Step 3: Add the exponents together
Next, we perform the addition of the exponents [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
- [tex]\(\frac{1}{2}\)[/tex] is equivalent to [tex]\(0.5\)[/tex]
- [tex]\(\frac{1}{4}\)[/tex] is equivalent to [tex]\(0.25\)[/tex]
When we add these together:
[tex]\[
\frac{1}{2} + \frac{1}{4} = 0.5 + 0.25 = 0.75
\][/tex]
So, the combined exponent is [tex]\(0.75\)[/tex].
### Step 4: Rewrite the expression with the simplified exponent
After adding the exponents, our expression becomes:
[tex]\[
x^{0.75}
\][/tex]
### Step 5: Convert the decimal exponent back to a fraction (optional)
Though [tex]\(0.75\)[/tex] is already a simplified form, it is often useful to convert it back to a fraction for clarity. The fraction equivalent of [tex]\(0.75\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Thus, the expression can also be written as:
[tex]\[
x^{\frac{3}{4}}
\][/tex]
### Conclusion
The expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] simplified in its simplest rational exponent form is:
[tex]\[
x^{0.75} \quad \text{or} \quad x^{\frac{3}{4}}
\][/tex]
Both forms are correct and represent the same simplified expression.
