Finite Math Examples

$x^{\frac{2}{7}} - 43 x^{\frac{1}{7}} = - 42$

Step 1

Add $42$ to both sides of the equation.

$x^{\frac{2}{7}} - 43 x^{\frac{1}{7}} + 42 = 0$

Step 2

Rewrite $x^{\frac{2}{7}}$ as ${(x^{\frac{1}{7}})}^{2}$ .

${(x^{\frac{1}{7}})}^{2} - 43 x^{\frac{1}{7}} + 42 = 0$

Step 3

Let $u = x^{\frac{1}{7}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{7}}$ .

$u^{2} - 43 u + 42 = 0$

Step 4

Factor $u^{2} - 43 u + 42$ using the AC method.

Tap for more steps...

Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $42$ and whose sum is $- 43$ .

$- 42, - 1$

Step 4.2

Write the factored form using these integers.

$(u - 42) (u - 1) = 0$

Step 5

Replace all occurrences of $u$ with $x^{\frac{1}{7}}$ .

$(x^{\frac{1}{7}} - 42) (x^{\frac{1}{7}} - 1) = 0$

Step 6

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

$x^{\frac{1}{7}} - 42 = 0$

$x^{\frac{1}{7}} - 1 = 0$

Step 7

Set $x^{\frac{1}{7}} - 42$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set $x^{\frac{1}{7}} - 42$ equal to $0$ .

$x^{\frac{1}{7}} - 42 = 0$

Step 7.2

Solve $x^{\frac{1}{7}} - 42 = 0$ for $x$ .

Tap for more steps...

Step 7.2.1

Add $42$ to both sides of the equation.

$x^{\frac{1}{7}} = 42$

Step 7.2.2

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{7}})}^{7} = 42^{7}$

Step 7.2.3

Simplify the exponent.

Tap for more steps...

Step 7.2.3.1

Simplify the left side.

Tap for more steps...

Step 7.2.3.1.1

Simplify ${(x^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 7.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 7.2.3.1.1.1.1

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

$x^{\frac{1}{7} \cdot 7} = 42^{7}$

Step 7.2.3.1.1.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 7.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = 42^{7}$

Step 7.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 42^{7}$

Step 7.2.3.1.1.2

Simplify.

$x = 42^{7}$

Step 7.2.3.2

Simplify the right side.

Tap for more steps...

Step 7.2.3.2.1

Raise $42$ to the power of $7$ .

$x = 230539333248$

Step 8

Set $x^{\frac{1}{7}} - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 8.1

Set $x^{\frac{1}{7}} - 1$ equal to $0$ .

$x^{\frac{1}{7}} - 1 = 0$

Step 8.2

Solve $x^{\frac{1}{7}} - 1 = 0$ for $x$ .

Tap for more steps...

Step 8.2.1

Add $1$ to both sides of the equation.

$x^{\frac{1}{7}} = 1$

Step 8.2.2

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{7}})}^{7} = 1^{7}$

Step 8.2.3

Simplify the exponent.

Tap for more steps...

Step 8.2.3.1

Simplify the left side.

Tap for more steps...

Step 8.2.3.1.1

Simplify ${(x^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 8.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 8.2.3.1.1.1.1

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

$x^{\frac{1}{7} \cdot 7} = 1^{7}$

Step 8.2.3.1.1.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 8.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{7} \cdot 7} = 1^{7}$

Step 8.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 1^{7}$

Step 8.2.3.1.1.2

Simplify.

$x = 1^{7}$

Step 8.2.3.2

Simplify the right side.

Tap for more steps...

Step 8.2.3.2.1

One to any power is one.

$x = 1$

Step 9

The final solution is all the values that make $(x^{\frac{1}{7}} - 42) (x^{\frac{1}{7}} - 1) = 0$ true.

$x = 230539333248, 1$