Kemarius Stewart $$ 3 / 20 / 25 $$ 42) Unit 3 Imvestigating Radical Functions Unit Review Advanced Algebra 1. White an equivalent expression for the following rational exponents: a. $$ x^{35} $$ b. ase c. $$ y^{07} $$ 2. Write an equivalent expression for the following radical expressions: a. $$ \sqrt[4]{16 x^{12} y^{10}} $$ b. $$ \sqrt[4]{625 x^{20} y^{18}} $$ c. $$ \sqrt[4]{256 x^{24} y^{22}} $$ 3. Simplify: a. $$ \left(\sqrt{a^{21}}\right)^{1 / 0} $$ b. $$ \left(\sqrt{x^{18}}\right)^{1 / 6} $$ c. $$ \left(\sqrt{y^{32}}\right)^{1 / 10} $$ 4. Solve: $$ x=\sqrt{x+6} $$

Question

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Let's solve the problems step by step.

### 1. Equivalent expressions for the following rational exponents:

a. $$ x^{35} $$  
This expression is already in its simplest form as a rational exponent.

b. $$ a^{\frac{1}{2}} $$  
This can be expressed as $$ \sqrt{a} $$.

c. $$ y^{\frac{7}{10}} $$  
This can be expressed as $$ \sqrt[10]{y^{7}} $$.

### 2. Equivalent expressions for the following radical expressions:

a. $$ \sqrt[4]{16 x^{12} y^{10}} $$  
We can simplify this as follows:
- $$ 16 = 2^4 $$
- $$ x^{12} = (x^3)^4 $$
- $$ y^{10} = (y^{2.5})^4 $$

Thus, we have:
$$
\sqrt[4]{16 x^{12} y^{10}} = \sqrt[4]{2^4} \cdot \sqrt[4]{(x^3)^4} \cdot \sqrt[4]{(y^{2.5})^4} = 2 x^3 y^{2.5}
$$

b. $$ \sqrt[4]{625 x^{20} y^{18}} $$  
We can simplify this as follows:
- $$ 625 = 25^2 = (5^2)^2 = 5^4 $$
- $$ x^{20} = (x^5)^4 $$
- $$ y^{18} = (y^{4.5})^4 $$

Thus, we have:
$$
\sqrt[4]{625 x^{20} y^{18}} = \sqrt[4]{5^4} \cdot \sqrt[4]{(x^5)^4} \cdot \sqrt[4]{(y^{4.5})^4} = 5 x^5 y^{4.5}
$$

c. $$ \sqrt[4]{256 x^{24} y^{22}} $$  
We can simplify this as follows:
- $$ 256 = 2^8 = (2^2)^4 = 4^4 $$
- $$ x^{24} = (x^6)^4 $$
- $$ y^{22} = (y^{5.5})^4 $$

Thus, we have:
$$
\sqrt[4]{256 x^{24} y^{22}} = \sqrt[4]{4^4} \cdot \sqrt[4]{(x^6)^4} \cdot \sqrt[4]{(y^{5.5})^4} = 4 x^6 y^{5.5}
$$

### 3. Simplify:

a. $$ \left(\sqrt{a^{21}}\right)^{1 / 0} $$  
Since $$ \frac{1}{0} $$ is undefined, this expression cannot be simplified further.

b. $$ \left(\sqrt{x^{18}}\right)^{1 / 6} $$  
This can be simplified as follows:
$$
\left(\sqrt{x^{18}}\right)^{1 / 6} = (x^{9})^{1 / 6} = x^{\frac{9}{6}} = x^{\frac{3}{2}}
$$

c. $$ \left(\sqrt{y^{32}}\right)^{1 / 10} $$  
This can be simplified as follows:
$$
\left(\sqrt{y^{32}}\right)^{1 / 10} = (y^{16})^{1 / 10} = y^{\frac{16}{10}} = y^{\frac{8}{5}}
$$

### 4. Solve: $$ x=\sqrt{x+6} $$

To solve this equation, we will square both sides to eliminate the square root:

$$
x^2 = x + 6
$$

Rearranging gives us:

$$
x^2 - x - 6 = 0
$$

Now, we can factor this quadratic equation:

$$
(x - 3)(x + 2) = 0
$$

Setting each factor to zero gives us:

$$
x - 3 = 0 \quad \Rightarrow \quad x = 3
$$
$$
x + 2 = 0 \quad \Rightarrow \quad x = -2
$$

Now, we need to check both solutions in the original equation:

1. For $$ x = 3 $$:
   $$
   3 = \sqrt{3 + 6} \quad \Rightarrow \quad 3 = \sqrt{9} \quad \Rightarrow \quad 3 = 3 \quad \text{(valid)}
   $$

2. For $$ x = -2 $$:
   $$
   -2 = \sqrt{-2 + 6} \quad \Rightarrow \quad -2 = \sqrt{4} \quad \Rightarrow \quad -2 = 2 \quad \text{(not valid)}
   $$

Thus, the only valid solution is $$ x = 3 $$.

### Summary of Answers:
1. a. $$ x^{35} $$  
   b. $$ \sqrt{a} $$  
   c. $$ \sqrt[10]{y^{7}} $$

2. a. $$ 2 x^3 y^{2.5} $$  
   b. $$ 5 x^5 y^{4.5} $$  
   c. $$ 4 x^6 y^{5.5} $$

3. a. Undefined  
   b. $$ x^{\frac{3}{2}} $$  
   c. $$ y^{\frac{8}{5}} $$

4. $$ x = 3 $$
1. a. $$ x^{35} $$ b. $$ \sqrt{a} $$ c. $$ \sqrt[10]{y^{7}} $$ 2. a. $$ 2 x^3 y^{2.5} $$ b. $$ 5 x^5 y^{4.5} $$ c. $$ 4 x^6 y^{5.5} $$ 3. a. Cannot simplify b. $$ x^{1.5} $$ c. $$ y^{1.6} $$ 4. $$ x = 3 $$

Kemarius Stewart

42)

Unit 3 Imvestigating Radical Functions Unit Review
Advanced Algebra

White an equivalent expression for the following rational exponents:
a.
b. ase
c.

Write an equivalent expression for the following radical expressions:
a.
b.
c.

Simplify:
a.
b.
c.

Solve:

Upstudy AI Solution

Answer

Solution

1. Equivalent expressions for the following rational exponents:

2. Equivalent expressions for the following radical expressions:

3. Simplify:

4. Solve:

Summary of Answers:

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Kemarius Stewart 42) Unit 3 Imvestigating Radical Functions Unit Review Advanced Algebra White an equivalent expression for the following rational exponents: a. b. ase c. Write an equivalent expression for the following radical expressions: a. b. c. Simplify: a. b. c. Solve: