Question

what is the radical form of 10^(3/2)? square root of 10 cubed cube root of 10 to the second power cube root of 10 squared square root of 10 to the third power

Explanation:

Step1: Recall exponent - radical rule

The rule for converting an exponential expression \(a^{\frac{m}{n}}\) to radical form is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (or equivalently \((\sqrt[n]{a})^{m}\)). Here, \(a = 10\), \(m=3\) and \(n = 2\).

Step2: Apply the rule to \(10^{\frac{3}{2}}\)

Using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), when \(a = 10\), \(m = 3\) and \(n=2\), we have \(10^{\frac{3}{2}}=\sqrt[2]{10^{3}}\), which is the square root of \(10\) cubed (or also can be written as \((\sqrt{10})^{3}\), but the form \(\sqrt{10^{3}}\) is the square root of \(10\) cubed). Also, note that \(\sqrt[2]{10^{3}}\) is the same as the square root of \(10\) to the third power (since \(10^{3}\) is \(10\) cubed or \(10\) to the third power). Let's analyze the options:
- Option 1: "Square root of 10 cubed" - \(10^{3}\) is \(10\) cubed, \(\sqrt{10^{3}}\) is square root of \(10\) cubed.
- Option 2: "Cube root of 10 to the second power" - This would be \(\sqrt[3]{10^{2}}\) which is \(10^{\frac{2}{3}}\), not \(10^{\frac{3}{2}}\).
- Option 3: "Cube root of 10 squared" - Same as option 2, \(\sqrt[3]{10^{2}}=10^{\frac{2}{3}}\).
- Option 4: "Square root of 10 to the third power" - \(10^{3}\) is \(10\) to the third power, \(\sqrt{10^{3}}\) is square root of \(10\) to the third power. Wait, but let's check the exponent rule again. \(10^{\frac{3}{2}}=\sqrt{10^{3}}\) (since \(n = 2\) is the index of the square root and \(m = 3\) is the power of the base inside the radical). So both "Square root of 10 cubed" and "Square root of 10 to the third power" are equivalent? Wait, no, "10 cubed" is \(10^{3}\) and "10 to the third power" is also \(10^{3}\). Wait, maybe there is a wording difference but let's check the exponent. Wait, maybe I made a mistake. Wait, \(10^{\frac{3}{2}}=\sqrt{10^{3}}\), which is square root of \(10^{3}\) (10 cubed) or square root of \(10\) to the third power. But let's check the options again. Wait, the fourth option is "Square root of 10 to the third power" and the first option is "Square root of 10 cubed". Since \(10^{3}\) is \(10\) cubed (or \(10\) to the third power), both are correct? Wait, no, maybe the options are written in different ways. Wait, let's re - express the exponent:

\(10^{\frac{3}{2}}=(10^{\frac{1}{2}})^{3}=\sqrt{10}^{3}\) or \(10^{\frac{3}{2}}=(10^{3})^{\frac{1}{2}}=\sqrt{10^{3}}\). So \(\sqrt{10^{3}}\) is the square root of \(10\) cubed (because \(10^{3}\) is \(10\) cubed) and also the square root of \(10\) to the third power (because \(10^{3}\) is \(10\) to the third power). But let's check the options:

The first option: "Square root of 10 cubed" - correct.

The fourth option: "Square root of 10 to the third power" - also correct? Wait, maybe there is a mistake in the options, but let's check the exponent again. Wait, \(10^{\frac{3}{2}}\): the numerator of the fraction in the exponent is the power of the base inside the radical (or the power of the radical expression), and the denominator is the index of the radical. So \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). So for \(10^{\frac{3}{2}}\), \(n = 2\) (square root), \(m = 3\) (so \(a^{m}=10^{3}\), which is \(10\) cubed). So \(\sqrt[2]{10^{3}}\) is the square root of \(10\) cubed. So the first option and the fourth option? Wait, no, "10 cubed" is \(10^{3}\) and "10 to the third power" is also \(10^{3}\). So both "Square root of 10 cubed" and "Square root of 10 to the third power" are equivalent. But let's check the options again. Wait, the first option is "Square root of 10 cubed" and the fourth is "Square root of 10 to the third power". But maybe in the context of the question, both are correct? Wait, no, let's re - evaluate. Wait, \(10^{\frac{3}{2}}\) can be written as \((\sqrt{10})^{3}\) or \(\sqrt{10^{3}}\). \(\sqrt{10^{3}}\) is the square root of \(10\) cubed (since \(10^{3}\) is \(10\) cubed) and also the square root of \(10\) to the third power (since \(10^{3}\) is \(10\) to the third power). But let's check the options:

- Option 1: "Square root of 10 cubed" - \(\sqrt{10^{3}}\)
- Option 4: "Square root of 10 to the third power" - \(\sqrt{10^{3}}\) (since \(10\) to the third power is \(10^{3}\))

Wait, but maybe the question considers "10 cubed" and "10 to the third power" as the same, so both option 1 and option 4? But that can't be. Wait, maybe I made a mistake. Wait, no, let's check the exponent rule again. The formula is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). So \(10^{\frac{3}{2}}=\sqrt[2]{10^{3}}\), which is the square root (index 2) of \(10\) raised to the 3rd power. So "square root of 10 to the third power" is \(\sqrt{10^{3}}\) and "square root of 10 cubed" is also \(\sqrt{10^{3}}\) (since \(10\) cubed is \(10^{3}\)). So both option 1 and option 4 are correct? But that's not possible in a multiple - choice question with one correct answer. Wait, maybe there is a typo or I misread the options. Let me check the options again:

- Green: Square root of 10 cubed
- Purple: Cube root of 10 to the second power
- Orange: Cube root of 10 squared
- Blue: Square root of 10 to the third power

Wait, no, the blue option is "Square root of 10 to the third power" and the green is "Square root of 10 cubed". But \(10^{3}\) is \(10\) cubed and \(10\) to the third power. So both green and blue? But that's not possible. Wait, maybe I made a mistake in the exponent rule. Wait, no, \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). So for \(10^{\frac{3}{2}}\), \(n = 2\) (square root), \(m = 3\) (so the power of \(a\) inside the radical is 3). So \(\sqrt{10^{3}}\) is the square root of \(10\) cubed (since \(10^{3}\) is \(10\) cubed) and also the square root of \(10\) to the third power (since \(10^{3}\) is \(10\) to the third power). So both green and blue options are correct? But that's unlikely. Wait, maybe the question has a mistake, but assuming that "10 cubed" and "10 to the third power" are the same, both are correct. But let's check the other options. The purple option is cube root of \(10\) squared (\(\sqrt[3]{10^{2}}\)) which is \(10^{\frac{2}{3}}\), not \(10^{\frac{3}{2}}\). The orange option is the same as purple. So the correct options are the green one ("Square root of 10 cubed") and the blue one ("Square root of 10 to the third power"). But maybe in the context of the question, they consider them as the same. But let's check the original problem again.

Wait, the problem is "What is the radical form of \(10^{\frac{3}{2}}\)?"

Using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), so \(10^{\frac{3}{2}}=\sqrt[2]{10^{3}}\), which is the square root of \(10\) cubed (since \(10^{3}\) is \(10\) cubed) and also the square root of \(10\) to the third power (since \(10^{3}\) is \(10\) to the third power). So both the first option (Square root of 10 cubed) and the fourth option (Square root of 10 to the third power) are correct. But maybe in the given options, the first one is "Square root of 10 cubed" and the fourth is "Square root of 10 to the third power". So we need to see which one is among the options. Wait, the first option is green: "Square root of 10 cubed", fourth is blue: "Square root of 10 to the third power". But maybe in the question, both are correct, but let's check the exponent again. Wait, \(10^{\frac{3}{2}}\) is equal to \(\sqrt{10^{3}}\) (square root of \(10\) cubed) and \((\sqrt{10})^{3}\) (square root of \(10\) cubed, because \((\sqrt{10})^{3}=\sqrt{10}\times\sqrt{10}\times\sqrt{10}=\sqrt{10\times10\times10}=\sqrt{10^{3}}\)). So the first option and the fourth option are correct. But maybe the question considers them as the same. So the correct options are:

A. Square root of 10 cubed

D. Square root of 10 to the third power

But let's check the options again. Wait, the user's options:

Green: Square root of 10 cubed

Purple: Cube root of 10 to the second power

Orange: Cube root of 10 squared

Blue: Square root of 10 to the third power

So both green and blue are correct. But maybe in the context of the question, they are considered the same. So the answer is the first option (Square root of 10 cubed) and the fourth option (Square root of 10 to the third power). But let's confirm with the exponent rule. Yes, \(10^{\frac{3}{2}}=\sqrt{10^{3}}\), which is the square root of \(10\) cubed (since \(10^{3}\) is \(10\) cubed) and also the square root of \(10\) to the third power (since \(10^{3}\) is \(10\) to the third power).

Answer:

A. Square root of 10 cubed, D. Square root of 10 to the third power

Wait, but maybe the question has a single correct answer. Wait, no, let's re - check. Wait, \(10^{\frac{3}{2}}\) can be written as \(\sqrt{10^{3}}\) (square root of \(10\) cubed) or \((\sqrt{10})^{3}\) (square root of \(10\) cubed). So "Square root of 10 cubed" is \(\sqrt{10^{3}}\) and "Square root of 10 to the third power" is also \(\sqrt{10^{3}}\) (since \(10\) to the third power is \(10^{3}\)). So both are correct. So the correct options are A and D.