Question

name kevin angel cash
5-1 lesson quiz
nth roots, radicals, and rational exponents

1. what are the real fourth roots of 256?

real fourth roots: (square) and (square)

1. select all expressions that are equivalent to (x^{\frac{9}{4}}).

(square) a. ((x^{3})^{\frac{1}{4}})
(square) b. (x^{\frac{8}{4}})
(square) c. ((x^{\frac{1}{4}})^{\frac{9}{2}})
(square) d. ((x^{\frac{1}{2}})^{3})
(square) e. (sqrt4{x^{9}})
(square) f. (sqrt4{x^{3}})

1. explain what (243^{\frac{2}{5}}) means, then evaluate it.

(243^{\frac{2}{5}}) is the (square) square
(square) cube
(square) fourth
(square) fifth root of 243 (square) squared
(square) cubed
(square) to the fourth power
(square) to the fifth power
its value is (square) 3
(square) 9
(square) 27
(square) 81

1. simplify the expression (sqrt{81m^{12}n^{4}}). assume all variables are positive.

(\boldsymbol{\text{a}}) (9m^{3}n)
(\boldsymbol{\text{b}}) (9m^{6}n^{2})
(\boldsymbol{\text{c}}) (3m^{3}n)
(\boldsymbol{\text{d}}) (3m^{8})

1. solve the equation (-4x^{3} = 32).

(x = square)

Explanation:

Question 1: Real fourth roots of 256

Step 1: Recall the definition of nth root

The real fourth roots of a number \( a \) are the numbers \( x \) such that \( x^4 = a \). So we need to find \( x \) where \( x^4 = 256 \).

Step 2: Find positive and negative roots

We know that \( 4^4 = 256 \) (since \( 4\times4\times4\times4 = 256 \)) and also \( (-4)^4 = 256 \) (because a negative number raised to an even power is positive).

We use the exponent rules \( (a^m)^n = a^{mn} \) and \( \sqrt[n]{a^m}=a^{\frac{m}{n}} \).

• Option A: \( (x^3)^{\frac{1}{2}} \). Using \( (a^m)^n = a^{mn} \), we get \( x^{3\times\frac{1}{2}} = x^{\frac{3}{2}} \). So A is equivalent.
• Option B: \( x^{\frac{6}{5}} \). The exponent is \( \frac{6}{5}

eq\frac{3}{2} \), so B is not equivalent.

• Option C: \( (x^{\frac{1}{2}})^{\frac{3}{2}} \). Using \( (a^m)^n = a^{mn} \), we get \( x^{\frac{1}{2}\times\frac{3}{2}} = x^{\frac{3}{4}}

eq x^{\frac{3}{2}} \). So C is not equivalent.

• Option D: \( (x^2)^{\frac{3}{4}} \). Using \( (a^m)^n = a^{mn} \), we get \( x^{2\times\frac{3}{4}} = x^{\frac{3}{2}} \). So D is equivalent.
• Option E: \( \sqrt[3]{x^4}=x^{\frac{4}{3}}

eq x^{\frac{3}{2}} \). So E is not equivalent.

• Option F: \( \sqrt[4]{x^3}=x^{\frac{3}{4}}

eq x^{\frac{3}{2}} \). So F is not equivalent.

Step 1: Interpret the exponent

The exponent \( \frac{2}{5} \) means we take the fifth root of 243 first (the denominator of the fraction in the exponent) and then square it (the numerator of the fraction in the exponent). So \( 243^{\frac{2}{5}} \) is the square of the fifth root of 243.

Step 2: Evaluate the fifth root of 243

We know that \( 3^5 = 243 \) (since \( 3\times3\times3\times3\times3 = 243 \)), so the fifth root of 243 is 3.

Step 3: Square the result

Now we square 3: \( 3^2 = 9 \).

Answer:

4 and -4

Question 2: Equivalent to \( x^{\frac{3}{2}} \)