Solved on Dec 14, 2023

Solve the equation with rational exponents: $(x-5)^{2/3} = 64$ . Select the correct choice: A. The solution set is $x = 10$ B. The solution set is the empty set

STEP 1

Assumptions
1. We are given the equation with a rational exponent: $(x-5)^{\frac{2}{3}} = 64$ .
2. We need to solve for the variable $x$ .
3. The exponent $\frac{2}{3}$ means the expression is squared and then the cube root is taken, or vice versa.

STEP 2

To eliminate the rational exponent, we can raise both sides of the equation to the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$\left((x-5)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 64^{\frac{3}{2}}$

STEP 3

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ .
$(x-5)^{\frac{2}{3} \times \frac{3}{2}} = (x-5)^1$

STEP 4

Simplify the left side of the equation.
$x-5 = 64^{\frac{3}{2}}$

STEP 5

Calculate the right side by first taking the square root of 64 and then cubing the result.
$64^{\frac{1}{2}} = 8$
$8^3 = 512$

STEP 6

Substitute the calculated value back into the equation.
$x-5 = 512$

STEP 7

Add 5 to both sides of the equation to solve for $x$ .
$x = 512 + 5$

STEP 8

Calculate the value of $x$ .
$x = 517$
The solution set is $x = 517$ .
A. The solution set is $\{517\}$ . B. The solution set is not the empty set.

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Solve the equation with rational exponents: $(x-5)^{2/3} = 64$ . Select the correct choice: A. The solution set is $x = 10$ B. The solution set is the empty set

STEP 1

Assumptions
1. We are given the equation with a rational exponent: $(x-5)^{\frac{2}{3}} = 64$ .
2. We need to solve for the variable $x$ .
3. The exponent $\frac{2}{3}$ means the expression is squared and then the cube root is taken, or vice versa.

STEP 2

To eliminate the rational exponent, we can raise both sides of the equation to the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$\left((x-5)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 64^{\frac{3}{2}}$

STEP 3

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ .
$(x-5)^{\frac{2}{3} \times \frac{3}{2}} = (x-5)^1$

STEP 4

Simplify the left side of the equation.
$x-5 = 64^{\frac{3}{2}}$

STEP 5

Calculate the right side by first taking the square root of 64 and then cubing the result.
$64^{\frac{1}{2}} = 8$
$8^3 = 512$

STEP 6

Substitute the calculated value back into the equation.
$x-5 = 512$

STEP 7

Add 5 to both sides of the equation to solve for $x$ .
$x = 512 + 5$

STEP 8

Calculate the value of $x$ .
$x = 517$
The solution set is $x = 517$ .
A. The solution set is $\{517\}$ . B. The solution set is not the empty set.

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Solve the equation with rational exponents: (x−5)2/3=64(x-5)^{2/3} = 64(x−5)2/3=64. Select the correct choice: A. The solution set is x=10x = 10x=10 B. The solution set is the empty set

STEP 1

Assumptions1. We are given the equation with a rational exponent: (x−5)23=64(x-5)^{\frac{2}{3}} = 64(x−5)32​=64.2. We need to solve for the variable xxx.3. The exponent 23\frac{2}{3}32​ means the expression is squared and then the cube root is taken, or vice versa.

STEP 2

To eliminate the rational exponent, we can raise both sides of the equation to the reciprocal of 23\frac{2}{3}32​, which is 32\frac{3}{2}23​.((x−5)23)32=6432\left((x-5)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 64^{\frac{3}{2}}((x−5)32​)23​=6423​

STEP 3

When we raise a power to a power, we multiply the exponents. The left side simplifies because 23×32=1\frac{2}{3} \times \frac{3}{2} = 132​×23​=1.(x−5)23×32=(x−5)1 (x-5)^{\frac{2}{3} \times \frac{3}{2}} = (x-5)^1 (x−5)32​×23​=(x−5)1

STEP 4

Simplify the left side of the equation.x−5=6432 x-5 = 64^{\frac{3}{2}} x−5=6423​

STEP 5

Calculate the right side by first taking the square root of 64 and then cubing the result.6412=8 64^{\frac{1}{2}} = 8 6421​=883=512 8^3 = 512 83=512

STEP 6

Substitute the calculated value back into the equation.x−5=512 x-5 = 512 x−5=512

STEP 7

Add 5 to both sides of the equation to solve for xxx.x=512+5 x = 512 + 5 x=512+5

STEP 8

Calculate the value of xxx.x=517 x = 517 x=517The solution set is x=517x = 517x=517.A. The solution set is {517} \{517\} {517}. B. The solution set is not the empty set.

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Solve the equation with rational exponents: $(x-5)^{2/3} = 64$ . Select the correct choice: A. The solution set is $x = 10$ B. The solution set is the empty set

Assumptions
1. We are given the equation with a rational exponent: $(x-5)^{\frac{2}{3}} = 64$ .
2. We need to solve for the variable $x$ .
3. The exponent $\frac{2}{3}$ means the expression is squared and then the cube root is taken, or vice versa.

To eliminate the rational exponent, we can raise both sides of the equation to the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$\left((x-5)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 64^{\frac{3}{2}}$

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ .
$(x-5)^{\frac{2}{3} \times \frac{3}{2}} = (x-5)^1$

Simplify the left side of the equation.
$x-5 = 64^{\frac{3}{2}}$

Calculate the right side by first taking the square root of 64 and then cubing the result.
$64^{\frac{1}{2}} = 8$
$8^3 = 512$

Substitute the calculated value back into the equation.
$x-5 = 512$

Add 5 to both sides of the equation to solve for $x$ .
$x = 512 + 5$

Calculate the value of $x$ .
$x = 517$
The solution set is $x = 517$ .
A. The solution set is $\{517\}$ . B. The solution set is not the empty set.