Calculus Examples

$y = x^{\frac{5}{3}} - 5 x^{\frac{2}{3}}$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = x^{\frac{5}{3}} - 5 x^{\frac{2}{3}}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $x^{\frac{5}{3}} - 5 x^{\frac{2}{3}} = 0$ .

$x^{\frac{5}{3}} - 5 x^{\frac{2}{3}} = 0$

Step 1.2.2

Find a common factor $x^{\frac{2}{3}}$ that is present in each term.

${(x^{\frac{2}{3}})}^{\frac{5}{2}} - 5 x^{\frac{2}{3}}$

Step 1.2.3

Substitute $u$ for $x^{\frac{2}{3}}$ .

${(u)}^{\frac{5}{2}} - 5 u = 0$

Step 1.2.4

Solve for $u$ .

Tap for more steps...

Step 1.2.4.1

Factor $u$ out of $u^{\frac{5}{2}} - 5 u$ .

Tap for more steps...

Step 1.2.4.1.1

Factor $u$ out of $u^{\frac{5}{2}}$ .

$u \cdot u^{\frac{3}{2}} - 5 u = 0$

Step 1.2.4.1.2

Factor $u$ out of $- 5 u$ .

$u \cdot u^{\frac{3}{2}} + u \cdot - 5 = 0$

Step 1.2.4.1.3

Factor $u$ out of $u \cdot u^{\frac{3}{2}} + u \cdot - 5$ .

$u (u^{\frac{3}{2}} - 5) = 0$

Step 1.2.4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$u^{\frac{3}{2}} - 5 = 0$

Step 1.2.4.3

Set $u$ equal to $0$ .

$u = 0$

Step 1.2.4.4

Set $u^{\frac{3}{2}} - 5$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 1.2.4.4.1

Set $u^{\frac{3}{2}} - 5$ equal to $0$ .

$u^{\frac{3}{2}} - 5 = 0$

Step 1.2.4.4.2

Solve $u^{\frac{3}{2}} - 5 = 0$ for $u$ .

Tap for more steps...

Step 1.2.4.4.2.1

Add $5$ to both sides of the equation.

$u^{\frac{3}{2}} = 5$

Step 1.2.4.4.2.2

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(u^{\frac{3}{2}})}^{\frac{2}{3}} = 5^{\frac{2}{3}}$

Step 1.2.4.4.2.3

Simplify the left side.

Tap for more steps...

Step 1.2.4.4.2.3.1

Simplify ${(u^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Tap for more steps...

Step 1.2.4.4.2.3.1.1

Multiply the exponents in ${(u^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Tap for more steps...

Step 1.2.4.4.2.3.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u^{\frac{3}{2} \cdot \frac{2}{3}} = 5^{\frac{2}{3}}$

Step 1.2.4.4.2.3.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.4.4.2.3.1.1.2.1

Cancel the common factor.

$u^{\frac{3}{2} \cdot \frac{2}{3}} = 5^{\frac{2}{3}}$

Step 1.2.4.4.2.3.1.1.2.2

Rewrite the expression.

$u^{\frac{1}{2} \cdot 2} = 5^{\frac{2}{3}}$

Step 1.2.4.4.2.3.1.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.4.4.2.3.1.1.3.1

Cancel the common factor.

$u^{\frac{1}{2} \cdot 2} = 5^{\frac{2}{3}}$

Step 1.2.4.4.2.3.1.1.3.2

Rewrite the expression.

$u^{1} = 5^{\frac{2}{3}}$

Step 1.2.4.4.2.3.1.2

Simplify.

$u = 5^{\frac{2}{3}}$

Step 1.2.4.5

The final solution is all the values that make $u (u^{\frac{3}{2}} - 5) = 0$ true.

$u = 0, 5^{\frac{2}{3}}$

Step 1.2.5

Substitute $x$ for $u$ .

$x^{\frac{2}{3}} = 0, 5^{\frac{2}{3}}$

Step 1.2.6

Solve for $x^{\frac{2}{3}} = 0$ for $x$ .

Tap for more steps...

Step 1.2.6.1

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 0^{\frac{3}{2}}$

Step 1.2.6.2

Simplify the exponent.

Tap for more steps...

Step 1.2.6.2.1

Simplify the left side.

Tap for more steps...

Step 1.2.6.2.1.1

Simplify ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 1.2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 1.2.6.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 0^{\frac{3}{2}}$

Step 1.2.6.2.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.6.2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 0^{\frac{3}{2}}$

Step 1.2.6.2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 0^{\frac{3}{2}}$

Step 1.2.6.2.1.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.6.2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 0^{\frac{3}{2}}$

Step 1.2.6.2.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 0^{\frac{3}{2}}$

Step 1.2.6.2.1.1.2

Simplify.

$x = \pm 0^{\frac{3}{2}}$

Step 1.2.6.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.6.2.2.1

Simplify $\pm 0^{\frac{3}{2}}$ .

Tap for more steps...

Step 1.2.6.2.2.1.1

Simplify the expression.

Tap for more steps...

Step 1.2.6.2.2.1.1.1

Rewrite $0$ as $0^{2}$ .

$x = \pm {(0^{2})}^{\frac{3}{2}}$

Step 1.2.6.2.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm 0^{2 (\frac{3}{2})}$

Step 1.2.6.2.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.6.2.2.1.2.1

Cancel the common factor.

$x = \pm 0^{2 (\frac{3}{2})}$

Step 1.2.6.2.2.1.2.2

Rewrite the expression.

$x = \pm 0^{3}$

Step 1.2.6.2.2.1.3

Raising $0$ to any positive power yields $0$ .

$x = \pm 0$

Step 1.2.6.2.2.1.4

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.7

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$x = 5$

Step 1.2.8

List all of the solutions.

$x = 0, 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (5, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = {(0)}^{\frac{5}{3}} - 5 {(0)}^{\frac{2}{3}}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 0^{\frac{5}{3}} - 5 {(0)}^{\frac{2}{3}}$

Step 2.2.2

Remove parentheses.

$y = 0^{\frac{5}{3}} - 5 \cdot 0^{\frac{2}{3}}$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{\frac{5}{3}} - 5 {(0)}^{\frac{2}{3}}$

Step 2.2.4

Simplify ${(0)}^{\frac{5}{3}} - 5 {(0)}^{\frac{2}{3}}$ .

Tap for more steps...

Step 2.2.4.1

Simplify each term.

Tap for more steps...

Step 2.2.4.1.1

Rewrite $0$ as $0^{3}$ .

$y = {(0^{3})}^{\frac{5}{3}} - 5 {(0)}^{\frac{2}{3}}$

Step 2.2.4.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = 0^{3 (\frac{5}{3})} - 5 {(0)}^{\frac{2}{3}}$

Step 2.2.4.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.4.1.3.1

Cancel the common factor.

$y = 0^{3 (\frac{5}{3})} - 5 {(0)}^{\frac{2}{3}}$

Step 2.2.4.1.3.2

Rewrite the expression.

$y = 0^{5} - 5 {(0)}^{\frac{2}{3}}$

Step 2.2.4.1.4

Raising $0$ to any positive power yields $0$ .

$y = 0 - 5 {(0)}^{\frac{2}{3}}$

Step 2.2.4.1.5

Rewrite $0$ as $0^{3}$ .

$y = 0 - 5 {(0^{3})}^{\frac{2}{3}}$

Step 2.2.4.1.6

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = 0 - 5 \cdot 0^{3 (\frac{2}{3})}$

Step 2.2.4.1.7

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.4.1.7.1

Cancel the common factor.

$y = 0 - 5 \cdot 0^{3 (\frac{2}{3})}$

Step 2.2.4.1.7.2

Rewrite the expression.

$y = 0 - 5 \cdot 0^{2}$

Step 2.2.4.1.8

Raising $0$ to any positive power yields $0$ .

$y = 0 - 5 \cdot 0$

Step 2.2.4.1.9

Multiply $- 5$ by $0$ .

$y = 0 + 0$

Step 2.2.4.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (5, 0)$

y-intercept(s): $(0, 0)$

Step 4