Math

Question Solve the equation x25=300x^{\frac{2}{5}} = 300 for the value of xx.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation x25=300x^{\frac{2}{5}}=300.
2. We need to solve for xx.

STEP 2

To solve for xx, 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}}

STEP 3

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

STEP 4

Now we need to calculate 30052300^{\frac{5}{2}}. This is equivalent to taking the square root of 300300 and then raising the result to the power of 55.
30052=(300)5300^{\frac{5}{2}} = \left(\sqrt{300}\right)^5

STEP 5

Calculate the square root of 300300.
300=100×3=100×3=103\sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3}

STEP 6

Now raise 10310\sqrt{3} to the power of 55.
(103)5=105×(3)5\left(10\sqrt{3}\right)^5 = 10^5 \times (\sqrt{3})^5

STEP 7

Since (3)5=(3)4×3(\sqrt{3})^5 = (\sqrt{3})^4 \times \sqrt{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}

STEP 8

Calculate 10510^5 and 323^2.
105=100,00010^5 = 100,000 32=93^2 = 9

STEP 9

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

STEP 10

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

STEP 11

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

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