Finite Math Examples

$\frac{7}{\sqrt{a}} \cdot a = 1$

Step 1

Solve for $\sqrt{a}$ .

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Step 1.1

Combine $\frac{7}{\sqrt{a}}$ and $a$ .

$\frac{7 a}{\sqrt{a}} = 1$

Step 1.2

Find the LCD of the terms in the equation.

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Step 1.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\sqrt{a}, 1$

Step 1.2.2

The LCM of one and any expression is the expression.

$\sqrt{a}$

Step 1.3

Multiply each term in $\frac{7 a}{\sqrt{a}} = 1$ by $\sqrt{a}$ to eliminate the fractions.

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Step 1.3.1

Multiply each term in $\frac{7 a}{\sqrt{a}} = 1$ by $\sqrt{a}$ .

$\frac{7 a}{\sqrt{a}} \sqrt{a} = 1 \sqrt{a}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $\sqrt{a}$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{7 a}{\sqrt{a}} \sqrt{a} = 1 \sqrt{a}$

Step 1.3.2.1.2

Rewrite the expression.

$7 a = 1 \sqrt{a}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Multiply $\sqrt{a}$ by $1$ .

$7 a = \sqrt{a}$

Step 1.4

Rewrite the equation as $\sqrt{a} = 7 a$ .

$\sqrt{a} = 7 a$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{a}}^{2} = {(7 a)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$ .

${(a^{\frac{1}{2}})}^{2} = {(7 a)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(a^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${(a^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

$a^{\frac{1}{2} \cdot 2} = {(7 a)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$a^{\frac{1}{2} \cdot 2} = {(7 a)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

$a^{1} = {(7 a)}^{2}$

Step 3.2.1.2

Simplify.

$a = {(7 a)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(7 a)}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $7 a$ .

$a = 7^{2} a^{2}$

Step 3.3.1.2

Raise $7$ to the power of $2$ .

$a = 49 a^{2}$

Step 4

Solve for $a$ .

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Step 4.1

Subtract $49 a^{2}$ from both sides of the equation.

$a - 49 a^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Let $u = a$ . Substitute $u$ for all occurrences of $a$ .

$u - 49 u^{2} = 0$

Step 4.2.2

Factor $u$ out of $u - 49 u^{2}$ .

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Step 4.2.2.1

Raise $u$ to the power of $1$ .

$u - 49 u^{2} = 0$

Step 4.2.2.2

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - 49 u^{2} = 0$

Step 4.2.2.3

Factor $u$ out of $- 49 u^{2}$ .

$u \cdot 1 + u (- 49 u) = 0$

Step 4.2.2.4

Factor $u$ out of $u \cdot 1 + u (- 49 u)$ .

$u (1 - 49 u) = 0$

Step 4.2.3

Replace all occurrences of $u$ with $a$ .

$a (1 - 49 a) = 0$

Step 4.3

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

$a = 0$

$1 - 49 a = 0$

Step 4.4

Set $a$ equal to $0$ .

$a = 0$

Step 4.5

Set $1 - 49 a$ equal to $0$ and solve for $a$ .

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Step 4.5.1

Set $1 - 49 a$ equal to $0$ .

$1 - 49 a = 0$

Step 4.5.2

Solve $1 - 49 a = 0$ for $a$ .

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Step 4.5.2.1

Subtract $1$ from both sides of the equation.

$- 49 a = - 1$

Step 4.5.2.2

Divide each term in $- 49 a = - 1$ by $- 49$ and simplify.

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Step 4.5.2.2.1

Divide each term in $- 49 a = - 1$ by $- 49$ .

$\frac{- 49 a}{- 49} = \frac{- 1}{- 49}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $- 49$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 49 a}{- 49} = \frac{- 1}{- 49}$

Step 4.5.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 1}{- 49}$

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Dividing two negative values results in a positive value.

$a = \frac{1}{49}$

Step 4.6

The final solution is all the values that make $a (1 - 49 a) = 0$ true.

$a = 0, \frac{1}{49}$

Step 5

Exclude the solutions that do not make $\frac{7}{\sqrt{a}} \cdot a = 1$ true.

$a = \frac{1}{49}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{1}{49}$

Decimal Form:

$a = 0.02040816 \dots$