Solved on Dec 12, 2023

Solve the equation $x^{\frac{2}{5}} = 300$ for the value of $x$ .

STEP 1

Assumptions
1. We are given the equation $x^{\frac{2}{5}}=300$ .
2. We need to solve for $x$ .

STEP 2

To solve for $x$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of the fractional exponent.
$\left(x^{\frac{2}{5}}\right)^{\frac{5}{2}} = 300^{\frac{5}{2}}$

STEP 3

On the left side of the equation, the exponent $\frac{2}{5}$ and the reciprocal exponent $\frac{5}{2}$ will cancel each other out, leaving us with $x$ .
$x = 300^{\frac{5}{2}}$

STEP 4

Now we need to calculate $300^{\frac{5}{2}}$ . This is equivalent to taking the square root of $300$ and then raising the result to the power of $5$ .
$300^{\frac{5}{2}} = \left(\sqrt{300}\right)^5$

STEP 5

Calculate the square root of $300$ .
$\sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3}$

STEP 6

Now raise $10\sqrt{3}$ to the power of $5$ .
$\left(10\sqrt{3}\right)^5 = 10^5 \times (\sqrt{3})^5$

STEP 7

Since $(\sqrt{3})^5 = (\sqrt{3})^4 \times \sqrt{3}$ , we can simplify this further.
$10^5 \times (\sqrt{3})^5 = 10^5 \times (3^2) \times \sqrt{3}$

STEP 8

Calculate $10^5$ and $3^2$ .
$10^5 = 100,000$ $3^2 = 9$

STEP 9

Now multiply these values together along with $\sqrt{3}$ .
$10^5 \times 9 \times \sqrt{3} = 100,000 \times 9 \times \sqrt{3}$

STEP 10

Calculate the product.
$100,000 \times 9 \times \sqrt{3} = 900,000 \times \sqrt{3}$

STEP 11

Since we cannot simplify $\sqrt{3}$ further without a calculator, we leave the solution in this form.
$x = 900,000 \times \sqrt{3}$
This is the solution to the given equation.

Was this helpful?

Solve the equation $x^{\frac{2}{5}} = 300$ for the value of $x$ .

STEP 1

Assumptions
1. We are given the equation $x^{\frac{2}{5}}=300$ .
2. We need to solve for $x$ .

STEP 2

To solve for $x$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of the fractional exponent.
$\left(x^{\frac{2}{5}}\right)^{\frac{5}{2}} = 300^{\frac{5}{2}}$

STEP 3

On the left side of the equation, the exponent $\frac{2}{5}$ and the reciprocal exponent $\frac{5}{2}$ will cancel each other out, leaving us with $x$ .
$x = 300^{\frac{5}{2}}$

STEP 4

Now we need to calculate $300^{\frac{5}{2}}$ . This is equivalent to taking the square root of $300$ and then raising the result to the power of $5$ .
$300^{\frac{5}{2}} = \left(\sqrt{300}\right)^5$

STEP 5

Calculate the square root of $300$ .
$\sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3}$

STEP 6

Now raise $10\sqrt{3}$ to the power of $5$ .
$\left(10\sqrt{3}\right)^5 = 10^5 \times (\sqrt{3})^5$

STEP 7

Since $(\sqrt{3})^5 = (\sqrt{3})^4 \times \sqrt{3}$ , we can simplify this further.
$10^5 \times (\sqrt{3})^5 = 10^5 \times (3^2) \times \sqrt{3}$

STEP 8

Calculate $10^5$ and $3^2$ .
$10^5 = 100,000$ $3^2 = 9$

STEP 9

Now multiply these values together along with $\sqrt{3}$ .
$10^5 \times 9 \times \sqrt{3} = 100,000 \times 9 \times \sqrt{3}$

STEP 10

Calculate the product.
$100,000 \times 9 \times \sqrt{3} = 900,000 \times \sqrt{3}$

STEP 11

Since we cannot simplify $\sqrt{3}$ further without a calculator, we leave the solution in this form.
$x = 900,000 \times \sqrt{3}$
This is the solution to the given equation.

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 x25=300x^{\frac{2}{5}} = 300x52​=300 for the value of xxx.

STEP 1

Assumptions1. We are given the equation x25=300x^{\frac{2}{5}}=300x52​=300.2. We need to solve for xxx.

STEP 2

To solve for xxx, we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of the fractional exponent.(x25)52=30052\left(x^{\frac{2}{5}}\right)^{\frac{5}{2}} = 300^{\frac{5}{2}}(x52​)25​=30025​

STEP 3

On the left side of the equation, the exponent 25\frac{2}{5}52​ and the reciprocal exponent 52\frac{5}{2}25​ will cancel each other out, leaving us with xxx.x=30052x = 300^{\frac{5}{2}}x=30025​

STEP 4

Now we need to calculate 30052300^{\frac{5}{2}}30025​. This is equivalent to taking the square root of 300300300 and then raising the result to the power of 555.30052=(300)5300^{\frac{5}{2}} = \left(\sqrt{300}\right)^530025​=(300​)5

STEP 5

Calculate the square root of 300300300.300=100×3=100×3=103\sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3}300​=100×3​=100​×3​=103​

STEP 6

Now raise 10310\sqrt{3}103​ to the power of 555.(103)5=105×(3)5\left(10\sqrt{3}\right)^5 = 10^5 \times (\sqrt{3})^5(103​)5=105×(3​)5

STEP 7

Since (3)5=(3)4×3(\sqrt{3})^5 = (\sqrt{3})^4 \times \sqrt{3}(3​)5=(3​)4×3​, we can simplify this further.105×(3)5=105×(32)×310^5 \times (\sqrt{3})^5 = 10^5 \times (3^2) \times \sqrt{3}105×(3​)5=105×(32)×3​

STEP 8

Calculate 10510^5105 and 323^232.105=100,00010^5 = 100,000105=100,000 32=93^2 = 932=9

STEP 9

Now multiply these values together along with 3\sqrt{3}3​.105×9×3=100,000×9×310^5 \times 9 \times \sqrt{3} = 100,000 \times 9 \times \sqrt{3}105×9×3​=100,000×9×3​

STEP 10

Calculate the product.100,000×9×3=900,000×3100,000 \times 9 \times \sqrt{3} = 900,000 \times \sqrt{3}100,000×9×3​=900,000×3​

STEP 11

Since we cannot simplify 3\sqrt{3}3​ further without a calculator, we leave the solution in this form.x=900,000×3x = 900,000 \times \sqrt{3}x=900,000×3​This is the solution to the given equation.

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 $x^{\frac{2}{5}} = 300$ for the value of $x$ .

Assumptions
1. We are given the equation $x^{\frac{2}{5}}=300$ .
2. We need to solve for $x$ .

To solve for $x$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal of the fractional exponent.
$\left(x^{\frac{2}{5}}\right)^{\frac{5}{2}} = 300^{\frac{5}{2}}$

On the left side of the equation, the exponent $\frac{2}{5}$ and the reciprocal exponent $\frac{5}{2}$ will cancel each other out, leaving us with $x$ .
$x = 300^{\frac{5}{2}}$

Now we need to calculate $300^{\frac{5}{2}}$ . This is equivalent to taking the square root of $300$ and then raising the result to the power of $5$ .
$300^{\frac{5}{2}} = \left(\sqrt{300}\right)^5$

Calculate the square root of $300$ .
$\sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3}$

Now raise $10\sqrt{3}$ to the power of $5$ .
$\left(10\sqrt{3}\right)^5 = 10^5 \times (\sqrt{3})^5$

Since $(\sqrt{3})^5 = (\sqrt{3})^4 \times \sqrt{3}$ , we can simplify this further.
$10^5 \times (\sqrt{3})^5 = 10^5 \times (3^2) \times \sqrt{3}$

Calculate $10^5$ and $3^2$ .
$10^5 = 100,000$ $3^2 = 9$

Now multiply these values together along with $\sqrt{3}$ .
$10^5 \times 9 \times \sqrt{3} = 100,000 \times 9 \times \sqrt{3}$

Calculate the product.
$100,000 \times 9 \times \sqrt{3} = 900,000 \times \sqrt{3}$

Since we cannot simplify $\sqrt{3}$ further without a calculator, we leave the solution in this form.
$x = 900,000 \times \sqrt{3}$
This is the solution to the given equation.