Solved on Jan 17, 2024

Find the value of $x$ in the equation $x=\sqrt[3]{(-245)^{-\frac{1}{5}}}$ rounded to the nearest hundredth.

STEP 1

Assumptions
1. We need to solve for $x$ in the equation $x=\sqrt[3]{(-245)^{-\frac{1}{5}}}$ .
2. The solution should be rounded to the nearest hundredth.

STEP 2

First, we need to simplify the expression inside the cube root. The expression $(-245)^{-\frac{1}{5}}$ represents the fifth root of $-245$ raised to the power of $-1$ .
$x = \sqrt[3]{(-245)^{-\frac{1}{5}}}$

STEP 3

Calculate the fifth root of $-245$ . The fifth root of a number is the number that, when raised to the power of 5, gives the original number. Since $-245$ is negative, its fifth root will also be negative.
$\sqrt[5]{-245} = -\left(\sqrt[5]{245}\right)$

STEP 4

Find the fifth root of $245$ . To do this, we can look for a number that, when raised to the power of 5, is close to $245$ .

STEP 5

By trial and error or estimation, we can determine that $3$ is a reasonable approximation for the fifth root of $245$ since $3^5 = 243$ .
$\sqrt[5]{245} \approx 3$

STEP 6

Now, we can use the approximation to rewrite the fifth root of $-245$ .
$\sqrt[5]{-245} \approx -3$

STEP 7

Next, we need to raise this result to the power of $-1$ to account for the negative exponent in the original expression.
$(-3)^{-1} = -\frac{1}{3}$

STEP 8

Now that we have simplified the expression inside the cube root, we can substitute it back into the original equation.
$x = \sqrt[3]{-\frac{1}{3}}$

STEP 9

To calculate the cube root of $-\frac{1}{3}$ , we need to find a number that, when raised to the power of 3, equals $-\frac{1}{3}$ .

STEP 10

By trial and error or estimation, we can determine that $-\frac{1}{\sqrt[3]{3}}$ is a reasonable approximation for the cube root of $-\frac{1}{3}$ .
$\sqrt[3]{-\frac{1}{3}} \approx -\frac{1}{\sqrt[3]{3}}$

STEP 11

Now, we can approximate the cube root of $3$ to find the value of $x$ .
$\sqrt[3]{3} \approx 1.442$

STEP 12

Using this approximation, we can now calculate the value of $x$ .
$x \approx -\frac{1}{1.442}$

STEP 13

Perform the division to find the value of $x$ .
$x \approx -\frac{1}{1.442} \approx -0.693$

STEP 14

Round the result to the nearest hundredth.
$x \approx -0.69$
The value of $x$ to the nearest hundredth is $-0.69$ .

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Find the value of $x$ in the equation $x=\sqrt[3]{(-245)^{-\frac{1}{5}}}$ rounded to the nearest hundredth.

STEP 1

Assumptions
1. We need to solve for $x$ in the equation $x=\sqrt[3]{(-245)^{-\frac{1}{5}}}$ .
2. The solution should be rounded to the nearest hundredth.

STEP 2

First, we need to simplify the expression inside the cube root. The expression $(-245)^{-\frac{1}{5}}$ represents the fifth root of $-245$ raised to the power of $-1$ .
$x = \sqrt[3]{(-245)^{-\frac{1}{5}}}$

STEP 3

Calculate the fifth root of $-245$ . The fifth root of a number is the number that, when raised to the power of 5, gives the original number. Since $-245$ is negative, its fifth root will also be negative.
$\sqrt[5]{-245} = -\left(\sqrt[5]{245}\right)$

STEP 4

Find the fifth root of $245$ . To do this, we can look for a number that, when raised to the power of 5, is close to $245$ .

STEP 5

By trial and error or estimation, we can determine that $3$ is a reasonable approximation for the fifth root of $245$ since $3^5 = 243$ .
$\sqrt[5]{245} \approx 3$

STEP 6

Now, we can use the approximation to rewrite the fifth root of $-245$ .
$\sqrt[5]{-245} \approx -3$

STEP 7

Next, we need to raise this result to the power of $-1$ to account for the negative exponent in the original expression.
$(-3)^{-1} = -\frac{1}{3}$

STEP 8

Now that we have simplified the expression inside the cube root, we can substitute it back into the original equation.
$x = \sqrt[3]{-\frac{1}{3}}$

STEP 9

To calculate the cube root of $-\frac{1}{3}$ , we need to find a number that, when raised to the power of 3, equals $-\frac{1}{3}$ .

STEP 10

By trial and error or estimation, we can determine that $-\frac{1}{\sqrt[3]{3}}$ is a reasonable approximation for the cube root of $-\frac{1}{3}$ .
$\sqrt[3]{-\frac{1}{3}} \approx -\frac{1}{\sqrt[3]{3}}$

STEP 11

Now, we can approximate the cube root of $3$ to find the value of $x$ .
$\sqrt[3]{3} \approx 1.442$

STEP 12

Using this approximation, we can now calculate the value of $x$ .
$x \approx -\frac{1}{1.442}$

STEP 13

Perform the division to find the value of $x$ .
$x \approx -\frac{1}{1.442} \approx -0.693$

STEP 14

Round the result to the nearest hundredth.
$x \approx -0.69$
The value of $x$ to the nearest hundredth is $-0.69$ .

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Find the value of xxx in the equation x=(−245)−153x=\sqrt[3]{(-245)^{-\frac{1}{5}}}x=3(−245)−51​​ rounded to the nearest hundredth.

STEP 1

Assumptions1. We need to solve for xxx in the equation x=(−245)−153x=\sqrt[3]{(-245)^{-\frac{1}{5}}}x=3(−245)−51​​.2. The solution should be rounded to the nearest hundredth.

STEP 2

First, we need to simplify the expression inside the cube root. The expression (−245)−15(-245)^{-\frac{1}{5}}(−245)−51​ represents the fifth root of −245-245−245 raised to the power of −1-1−1.x=(−245)−153x = \sqrt[3]{(-245)^{-\frac{1}{5}}}x=3(−245)−51​​

STEP 3

STEP 4

Find the fifth root of 245245245. To do this, we can look for a number that, when raised to the power of 5, is close to 245245245.

STEP 5

By trial and error or estimation, we can determine that 333 is a reasonable approximation for the fifth root of 245245245 since 35=2433^5 = 24335=243.2455≈3\sqrt[5]{245} \approx 35245​≈3

STEP 6

Now, we can use the approximation to rewrite the fifth root of −245-245−245.−2455≈−3\sqrt[5]{-245} \approx -35−245​≈−3

STEP 7

Next, we need to raise this result to the power of −1-1−1 to account for the negative exponent in the original expression.(−3)−1=−13(-3)^{-1} = -\frac{1}{3}(−3)−1=−31​

STEP 8

Now that we have simplified the expression inside the cube root, we can substitute it back into the original equation.x=−133x = \sqrt[3]{-\frac{1}{3}}x=3−31​​

STEP 9

To calculate the cube root of −13-\frac{1}{3}−31​, we need to find a number that, when raised to the power of 3, equals −13-\frac{1}{3}−31​.

STEP 10

By trial and error or estimation, we can determine that −133-\frac{1}{\sqrt[3]{3}}−33​1​ is a reasonable approximation for the cube root of −13-\frac{1}{3}−31​.−133≈−133\sqrt[3]{-\frac{1}{3}} \approx -\frac{1}{\sqrt[3]{3}}3−31​​≈−33​1​

STEP 11

Now, we can approximate the cube root of 333 to find the value of xxx.33≈1.442\sqrt[3]{3} \approx 1.44233​≈1.442

STEP 12

Using this approximation, we can now calculate the value of xxx.x≈−11.442x \approx -\frac{1}{1.442}x≈−1.4421​

STEP 13

Perform the division to find the value of xxx.x≈−11.442≈−0.693x \approx -\frac{1}{1.442} \approx -0.693x≈−1.4421​≈−0.693

STEP 14

Round the result to the nearest hundredth.x≈−0.69x \approx -0.69x≈−0.69The value of xxx to the nearest hundredth is −0.69-0.69−0.69.

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Find the value of $x$ in the equation $x=\sqrt[3]{(-245)^{-\frac{1}{5}}}$ rounded to the nearest hundredth.

Assumptions
1. We need to solve for $x$ in the equation $x=\sqrt[3]{(-245)^{-\frac{1}{5}}}$ .
2. The solution should be rounded to the nearest hundredth.

First, we need to simplify the expression inside the cube root. The expression $(-245)^{-\frac{1}{5}}$ represents the fifth root of $-245$ raised to the power of $-1$ .
$x = \sqrt[3]{(-245)^{-\frac{1}{5}}}$

Find the fifth root of $245$ . To do this, we can look for a number that, when raised to the power of 5, is close to $245$ .

By trial and error or estimation, we can determine that $3$ is a reasonable approximation for the fifth root of $245$ since $3^5 = 243$ .
$\sqrt[5]{245} \approx 3$

Now, we can use the approximation to rewrite the fifth root of $-245$ .
$\sqrt[5]{-245} \approx -3$

Next, we need to raise this result to the power of $-1$ to account for the negative exponent in the original expression.
$(-3)^{-1} = -\frac{1}{3}$

Now that we have simplified the expression inside the cube root, we can substitute it back into the original equation.
$x = \sqrt[3]{-\frac{1}{3}}$

To calculate the cube root of $-\frac{1}{3}$ , we need to find a number that, when raised to the power of 3, equals $-\frac{1}{3}$ .

By trial and error or estimation, we can determine that $-\frac{1}{\sqrt[3]{3}}$ is a reasonable approximation for the cube root of $-\frac{1}{3}$ .
$\sqrt[3]{-\frac{1}{3}} \approx -\frac{1}{\sqrt[3]{3}}$

Now, we can approximate the cube root of $3$ to find the value of $x$ .
$\sqrt[3]{3} \approx 1.442$

Using this approximation, we can now calculate the value of $x$ .
$x \approx -\frac{1}{1.442}$

Perform the division to find the value of $x$ .
$x \approx -\frac{1}{1.442} \approx -0.693$

Round the result to the nearest hundredth.
$x \approx -0.69$
The value of $x$ to the nearest hundredth is $-0.69$ .