Basic Math Examples

$9^{f} \cdot 3^{f^{2}} = 81^{2}$

Step 1

Rewrite $9$ as $3^{2}$ .

${(3^{2})}^{f} \cdot 3^{f^{2}} = 81^{2}$

Step 2

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

$3^{2 f} \cdot 3^{f^{2}} = 81^{2}$

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3^{2 f + f^{2}} = 81^{2}$

Step 4

Create equivalent expressions in the equation that all have equal bases.

$3^{2 f + f^{2}} = 3^{4 \cdot 2}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 f + f^{2} = 4 \cdot 2$

Step 6

Solve for $f$ .

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Step 6.1

Multiply $4$ by $2$ .

$2 f + f^{2} = 8$

Step 6.2

Subtract $8$ from both sides of the equation.

$2 f + f^{2} - 8 = 0$

Step 6.3

Factor the left side of the equation.

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Step 6.3.1

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

$2 u + u^{2} - 8 = 0$

Step 6.3.2

Factor $2 u + u^{2} - 8$ using the AC method.

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Step 6.3.2.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 6.3.2.2

Write the factored form using these integers.

$(u - 2) (u + 4) = 0$

Step 6.3.3

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

$(f - 2) (f + 4) = 0$

Step 6.4

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

$f - 2 = 0$

$f + 4 = 0$

Step 6.5

Set $f - 2$ equal to $0$ and solve for $f$ .

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Step 6.5.1

Set $f - 2$ equal to $0$ .

$f - 2 = 0$

Step 6.5.2

Add $2$ to both sides of the equation.

$f = 2$

Step 6.6

Set $f + 4$ equal to $0$ and solve for $f$ .

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Step 6.6.1

Set $f + 4$ equal to $0$ .

$f + 4 = 0$

Step 6.6.2

Subtract $4$ from both sides of the equation.

$f = - 4$

Step 6.7

The final solution is all the values that make $(f - 2) (f + 4) = 0$ true.

$f = 2, - 4$