Algebra Examples

$3^{\frac{1}{3}}$ , $3^{\frac{2}{3}}$ , $3^{1}$ , $3^{\frac{4}{3}}$

Step 1

Evaluate the exponent.

$3^{\frac{1}{3}}, 3^{\frac{2}{3}}, 3, 3^{\frac{4}{3}}$

Step 2

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $3^{\frac{1}{3}}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = 3^{\frac{1}{3}}$

Step 3

This is the form of a geometric sequence.

$a_{n} = a_{1} r^{n - 1}$

Step 4

Substitute in the values of $a_{1} = 3^{\frac{1}{3}}$ and $r = 3^{\frac{1}{3}}$ .

$a_{n} = 3^{\frac{1}{3}} {(3^{\frac{1}{3}})}^{n - 1}$

Step 5

Multiply the exponents in ${(3^{\frac{1}{3}})}^{n - 1}$ .

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Step 5.1

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

$a_{n} = 3^{\frac{1}{3}} \cdot 3^{\frac{1}{3} (n - 1)}$

Step 5.2

Apply the distributive property.

$a_{n} = 3^{\frac{1}{3}} \cdot 3^{\frac{1}{3} n + \frac{1}{3} \cdot - 1}$

Step 5.3

Combine $\frac{1}{3}$ and $n$ .

$a_{n} = 3^{\frac{1}{3}} \cdot 3^{\frac{n}{3} + \frac{1}{3} \cdot - 1}$

Step 5.4

Combine $\frac{1}{3}$ and $- 1$ .

$a_{n} = 3^{\frac{1}{3}} \cdot 3^{\frac{n}{3} + \frac{- 1}{3}}$

Step 5.5

Move the negative in front of the fraction.

$a_{n} = 3^{\frac{1}{3}} \cdot 3^{\frac{n}{3} - \frac{1}{3}}$

Step 6

Multiply $3^{\frac{1}{3}}$ by $3^{\frac{n}{3} - \frac{1}{3}}$ by adding the exponents.

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Step 6.1

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

$a_{n} = 3^{\frac{1}{3} + \frac{n}{3} - \frac{1}{3}}$

Step 6.2

Combine the opposite terms in $\frac{1}{3} + \frac{n}{3} - \frac{1}{3}$ .

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Step 6.2.1

Combine the numerators over the common denominator.

$a_{n} = 3^{\frac{n}{3} + \frac{1 - 1}{3}}$

Step 6.2.2

Subtract $1$ from $1$ .

$a_{n} = 3^{\frac{n}{3} + \frac{0}{3}}$

Step 6.2.3

Divide $0$ by $3$ .

$a_{n} = 3^{\frac{n}{3} + 0}$

Step 6.2.4

Add $\frac{n}{3}$ and $0$ .

$a_{n} = 3^{\frac{n}{3}}$