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Rewrite in simplest rational exponent form [tex]\sqrt{x} \cdot \sqrt[4]{x}[/tex]. Show each step of your process.

Step 1: Express each root as a rational exponent.

[tex]\[
\sqrt{x} = x^{\frac{1}{2}}
\][/tex]
[tex]\[
\sqrt[4]{x} = x^{\frac{1}{4}}
\][/tex]

Step 2: Multiply the exponents.

[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} = x^{\frac{1}{2} + \frac{1}{4}}
\][/tex]

Step 3: Add the exponents.

[tex]\[
\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}
\][/tex]

Step 4: Write the final expression.

[tex]\[
x^{\frac{3}{4}}
\][/tex]

Therefore, the simplest rational exponent form of [tex]\sqrt{x} \cdot \sqrt[4]{x}[/tex] is [tex]x^{\frac{3}{4}}[/tex].

Sagot :

GinnyAnsweridn
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.
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