Solved on Feb 17, 2024

Solve the equation with rational exponents: $(x-5)^{2/3} = 25$ .

STEP 1

Assumptions
1. We are given the equation with a rational exponent: $(x-5)^{2/3} = 25$ .
2. We need to solve for $x$ .
3. Rational exponents follow the rules of exponents and can be manipulated using algebraic principles.

STEP 2

To remove the rational exponent, we can raise both sides of the equation to the reciprocal of the exponent. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ .
$\left((x-5)^{2/3}\right)^{3/2} = 25^{3/2}$

STEP 3

When we raise a power to a power, we multiply the exponents. Here, $\frac{2}{3} \times \frac{3}{2} = 1$ , so the left side of the equation simplifies to $x-5$ .
$x-5 = 25^{3/2}$

STEP 4

We now need to calculate $25^{3/2}$ . The exponent $\frac{3}{2}$ means that we should take the square root of 25 and then raise it to the power of 3.
$25^{3/2} = (25^{1/2})^3 = (5)^3$

STEP 5

Calculate the value of $5^3$ .
$5^3 = 5 \times 5 \times 5 = 125$

STEP 6

Substitute the calculated value back into the equation.
$x-5 = 125$

STEP 7

To solve for $x$ , add 5 to both sides of the equation.
$x = 125 + 5$

STEP 8

Calculate the value of $x$ .
$x = 130$
The solution to the equation $(x-5)^{2/3} = 25$ is $x = 130$ .

Was this helpful?

Solve the equation with rational exponents: $(x-5)^{2/3} = 25$ .

STEP 1

Assumptions
1. We are given the equation with a rational exponent: $(x-5)^{2/3} = 25$ .
2. We need to solve for $x$ .
3. Rational exponents follow the rules of exponents and can be manipulated using algebraic principles.

STEP 2

To remove the rational exponent, we can raise both sides of the equation to the reciprocal of the exponent. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ .
$\left((x-5)^{2/3}\right)^{3/2} = 25^{3/2}$

STEP 3

When we raise a power to a power, we multiply the exponents. Here, $\frac{2}{3} \times \frac{3}{2} = 1$ , so the left side of the equation simplifies to $x-5$ .
$x-5 = 25^{3/2}$

STEP 4

We now need to calculate $25^{3/2}$ . The exponent $\frac{3}{2}$ means that we should take the square root of 25 and then raise it to the power of 3.
$25^{3/2} = (25^{1/2})^3 = (5)^3$

STEP 5

Calculate the value of $5^3$ .
$5^3 = 5 \times 5 \times 5 = 125$

STEP 6

Substitute the calculated value back into the equation.
$x-5 = 125$

STEP 7

To solve for $x$ , add 5 to both sides of the equation.
$x = 125 + 5$

STEP 8

Calculate the value of $x$ .
$x = 130$
The solution to the equation $(x-5)^{2/3} = 25$ is $x = 130$ .

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the equation with rational exponents: (x−5)2/3=25(x-5)^{2/3} = 25(x−5)2/3=25.

STEP 1

Assumptions1. We are given the equation with a rational exponent: (x−5)2/3=25(x-5)^{2/3} = 25(x−5)2/3=25.2. We need to solve for xxx.3. Rational exponents follow the rules of exponents and can be manipulated using algebraic principles.

STEP 2

To remove the rational exponent, we can raise both sides of the equation to the reciprocal of the exponent. The reciprocal of 23\frac{2}{3}32​ is 32\frac{3}{2}23​.((x−5)2/3)3/2=253/2 \left((x-5)^{2/3}\right)^{3/2} = 25^{3/2} ((x−5)2/3)3/2=253/2

STEP 3

When we raise a power to a power, we multiply the exponents. Here, 23×32=1\frac{2}{3} \times \frac{3}{2} = 132​×23​=1, so the left side of the equation simplifies to x−5x-5x−5.x−5=253/2 x-5 = 25^{3/2} x−5=253/2

STEP 4

We now need to calculate 253/225^{3/2}253/2. The exponent 32\frac{3}{2}23​ means that we should take the square root of 25 and then raise it to the power of 3.253/2=(251/2)3=(5)3 25^{3/2} = (25^{1/2})^3 = (5)^3 253/2=(251/2)3=(5)3

STEP 5

Calculate the value of 535^353.53=5×5×5=125 5^3 = 5 \times 5 \times 5 = 125 53=5×5×5=125

STEP 6

Substitute the calculated value back into the equation.x−5=125 x-5 = 125 x−5=125

STEP 7

To solve for xxx, add 5 to both sides of the equation.x=125+5 x = 125 + 5 x=125+5

STEP 8

Calculate the value of xxx.x=130 x = 130 x=130The solution to the equation (x−5)2/3=25(x-5)^{2/3} = 25(x−5)2/3=25 is x=130x = 130x=130.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the equation with rational exponents: $(x-5)^{2/3} = 25$ .

Assumptions
1. We are given the equation with a rational exponent: $(x-5)^{2/3} = 25$ .
2. We need to solve for $x$ .
3. Rational exponents follow the rules of exponents and can be manipulated using algebraic principles.

To remove the rational exponent, we can raise both sides of the equation to the reciprocal of the exponent. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ .
$\left((x-5)^{2/3}\right)^{3/2} = 25^{3/2}$

When we raise a power to a power, we multiply the exponents. Here, $\frac{2}{3} \times \frac{3}{2} = 1$ , so the left side of the equation simplifies to $x-5$ .
$x-5 = 25^{3/2}$

We now need to calculate $25^{3/2}$ . The exponent $\frac{3}{2}$ means that we should take the square root of 25 and then raise it to the power of 3.
$25^{3/2} = (25^{1/2})^3 = (5)^3$

Calculate the value of $5^3$ .
$5^3 = 5 \times 5 \times 5 = 125$

Substitute the calculated value back into the equation.
$x-5 = 125$

To solve for $x$ , add 5 to both sides of the equation.
$x = 125 + 5$

Calculate the value of $x$ .
$x = 130$
The solution to the equation $(x-5)^{2/3} = 25$ is $x = 130$ .