Questions: (a) Using exponential notation, we can write the cube root of 10 as . (b) Using radicals, we can write 10 to the power of 1/2 as . (c) Is there a difference between the square root of 10 squared and (the square root of 10) squared? Yes No

(a) Using exponential notation, we can write the cube root of 10 as .
(b) Using radicals, we can write 10 to the power of 1/2 as .
(c) Is there a difference between the square root of 10 squared and (the square root of 10) squared?
Yes
No

Transcript text: (a) Using exponential notation, we can write $\sqrt[3]{10}$ as $\square$ . (b) Using radicals, we can write $10^{1 / 2}$ as $\square$ . (c) Is there a difference between $\sqrt{10^{2}}$ and $(\sqrt{10})^{2}$ ? Yes No

Solution

Solution Steps

Solution Approach

(a) To express the cube root of 10 using exponential notation, we use the fact that the cube root of a number $ x $ can be written as $ x^{1/3} $.

(b) To express $ 10^{1/2} $ using radicals, we use the fact that $ x^{1/2} $ is the same as the square root of $ x $.

(c) To determine if there is a difference between $ \sqrt{10^2} $ and $ (\sqrt{10})^2 $, we need to simplify both expressions and compare the results.

Step 1: Expressing $\sqrt[3]{10}$ in Exponential Notation

Using exponential notation, we can write the cube root of 10 as: \[ \sqrt[3]{10} = 10^{1/3} \approx 2.1544 \]

Step 2: Expressing $10^{1/2}$ Using Radicals

Using radicals, we can express $10^{1/2}$ as: \[ 10^{1/2} = \sqrt{10} \approx 3.1623 \]

Step 3: Comparing $\sqrt{10^2}$ and $(\sqrt{10})^2$

We simplify both expressions: \[ \sqrt{10^2} = 10 \] \[ (\sqrt{10})^2 = 10 \] However, due to floating-point precision, we find: \[ \sqrt{10^2} = 10.0 \quad \text{and} \quad (\sqrt{10})^2 \approx 10.0000 \] Thus, while they are mathematically equal, the comparison shows a slight difference in their computed values.