Precalculus Examples

$7 + 42 \cdot 3^{2 - 3 a} = 14 \cdot 3^{- 2 a} + 7$

Step 1

Subtract $14 \cdot 3^{- 2 a}$ from both sides of the equation.

$7 + 42 \cdot 3^{2 - 3 a} - 14 \cdot 3^{- 2 a} = 7$

Step 2

Rewrite $3^{2 - 3 a}$ as $3^{2} \cdot 3^{- 3 a}$ .

$7 + 42 \cdot (3^{2} \cdot 3^{- 3 a}) - 14 \cdot 3^{- 2 a} = 7$

Step 3

Rewrite $3^{- 3 a}$ as exponentiation.

$7 + 42 \cdot (3^{2} \cdot {(3^{a})}^{- 3}) - 14 \cdot 3^{- 2 a} = 7$

Step 4

Rewrite $3^{- 2 a}$ as exponentiation.

$7 + 42 \cdot (3^{2} \cdot {(3^{a})}^{- 3}) - 14 \cdot {(3^{a})}^{- 2} = 7$

Step 5

Multiply $- 14$ by ${3^{a}}^{- 2}$ .

$7 + 42 \cdot (3^{2} {(3^{a})}^{- 3}) - 14 {(3^{a})}^{- 2} = 7$

Step 6

Substitute $u$ for $3^{a}$ .

$7 + 42 \cdot (3^{2} u^{- 3}) - 14 u^{- 2} = 7$

Step 7

Simplify each term.

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Step 7.1

Raise $3$ to the power of $2$ .

$7 + 42 \cdot (9 u^{- 3}) - 14 u^{- 2} = 7$

Step 7.2

Multiply $42$ by $9$ .

$7 + 378 u^{- 3} - 14 u^{- 2} = 7$

Step 7.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$7 + 378 (\frac{1}{u^{3}}) - 14 u^{- 2} = 7$

Step 7.4

Combine $378$ and $\frac{1}{u^{3}}$ .

$7 + \frac{378}{u^{3}} - 14 u^{- 2} = 7$

Step 7.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$7 + \frac{378}{u^{3}} - 14 \frac{1}{u^{2}} = 7$

Step 7.6

Combine $- 14$ and $\frac{1}{u^{2}}$ .

$7 + \frac{378}{u^{3}} + \frac{- 14}{u^{2}} = 7$

Step 7.7

Move the negative in front of the fraction.

$7 + \frac{378}{u^{3}} - \frac{14}{u^{2}} = 7$

Step 8

Solve for $u$ .

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Step 8.1

Find the LCD of the terms in the equation.

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Step 8.1.1

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

$1, u^{3}, u^{2}, 1$

Step 8.1.2

Since $1, u^{3}, u^{2}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $u^{3}, u^{2}$ .

Step 8.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 8.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 8.1.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 8.1.6

The factors for $u^{3}$ are $u \cdot u \cdot u$ , which is $u$ multiplied by each other $3$ times.

$u^{3} = u \cdot u \cdot u$

$u$ occurs $3$ times.

Step 8.1.7

The factors for $u^{2}$ are $u \cdot u$ , which is $u$ multiplied by each other $2$ times.

$u^{2} = u \cdot u$

$u$ occurs $2$ times.

Step 8.1.8

The LCM of $u^{3}, u^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$u \cdot u \cdot u$

Step 8.1.9

Simplify $u \cdot u \cdot u$ .

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Step 8.1.9.1

Multiply $u$ by $u$ .

$u^{2} \cdot u$

Step 8.1.9.2

Multiply $u^{2}$ by $u$ by adding the exponents.

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Step 8.1.9.2.1

Multiply $u^{2}$ by $u$ .

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Step 8.1.9.2.1.1

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

$u^{2} \cdot u^{1}$

Step 8.1.9.2.1.2

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

$u^{2 + 1}$

Step 8.1.9.2.2

Add $2$ and $1$ .

$u^{3}$

Step 8.2

Multiply each term in $7 + \frac{378}{u^{3}} - \frac{14}{u^{2}} = 7$ by $u^{3}$ to eliminate the fractions.

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Step 8.2.1

Multiply each term in $7 + \frac{378}{u^{3}} - \frac{14}{u^{2}} = 7$ by $u^{3}$ .

$7 u^{3} + \frac{378}{u^{3}} u^{3} - \frac{14}{u^{2}} u^{3} = 7 u^{3}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Simplify each term.

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Step 8.2.2.1.1

Cancel the common factor of $u^{3}$ .

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Step 8.2.2.1.1.1

Cancel the common factor.

$7 u^{3} + \frac{378}{u^{3}} u^{3} - \frac{14}{u^{2}} u^{3} = 7 u^{3}$

Step 8.2.2.1.1.2

Rewrite the expression.

$7 u^{3} + 378 - \frac{14}{u^{2}} u^{3} = 7 u^{3}$

Step 8.2.2.1.2

Cancel the common factor of $u^{2}$ .

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Step 8.2.2.1.2.1

Move the leading negative in $- \frac{14}{u^{2}}$ into the numerator.

$7 u^{3} + 378 + \frac{- 14}{u^{2}} u^{3} = 7 u^{3}$

Step 8.2.2.1.2.2

Factor $u^{2}$ out of $u^{3}$ .

$7 u^{3} + 378 + \frac{- 14}{u^{2}} (u^{2} u) = 7 u^{3}$

Step 8.2.2.1.2.3

Cancel the common factor.

$7 u^{3} + 378 + \frac{- 14}{u^{2}} (u^{2} u) = 7 u^{3}$

Step 8.2.2.1.2.4

Rewrite the expression.

$7 u^{3} + 378 - 14 u = 7 u^{3}$

Step 8.3

Solve the equation.

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Step 8.3.1

Move all terms containing $u$ to the left side of the equation.

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Step 8.3.1.1

Subtract $7 u^{3}$ from both sides of the equation.

$7 u^{3} + 378 - 14 u - 7 u^{3} = 0$

Step 8.3.1.2

Combine the opposite terms in $7 u^{3} + 378 - 14 u - 7 u^{3}$ .

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Step 8.3.1.2.1

Subtract $7 u^{3}$ from $7 u^{3}$ .

$378 - 14 u + 0 = 0$

Step 8.3.1.2.2

Add $378 - 14 u$ and $0$ .

$378 - 14 u = 0$

Step 8.3.2

Subtract $378$ from both sides of the equation.

$- 14 u = - 378$

Step 8.3.3

Divide each term in $- 14 u = - 378$ by $- 14$ and simplify.

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Step 8.3.3.1

Divide each term in $- 14 u = - 378$ by $- 14$ .

$\frac{- 14 u}{- 14} = \frac{- 378}{- 14}$

Step 8.3.3.2

Simplify the left side.

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Step 8.3.3.2.1

Cancel the common factor of $- 14$ .

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Step 8.3.3.2.1.1

Cancel the common factor.

$\frac{- 14 u}{- 14} = \frac{- 378}{- 14}$

Step 8.3.3.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 378}{- 14}$

Step 8.3.3.3

Simplify the right side.

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Step 8.3.3.3.1

Divide $- 378$ by $- 14$ .

$u = 27$

Step 9

Substitute $27$ for $u$ in $u = 3^{a}$ .

$27 = 3^{a}$

Step 10

Solve $27 = 3^{a}$ .

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Step 10.1

Rewrite the equation as $3^{a} = 27$ .

$3^{a} = 27$

Step 10.2

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

$3^{a} = 3^{3}$

Step 10.3

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

$a = 3$