Basic Math Examples

$3.3 \cdot 10^{5} = (3 \cdot 10^{2}) (1.1 \cdot 10^{n})$

Step 1

Rewrite the equation as $3 \cdot 10^{2} \cdot 1.1 \cdot 10^{n} = 3.3 \cdot 10^{5}$ .

$3 \cdot 10^{2} \cdot 1.1 \cdot 10^{n} = 3.3 \cdot 10^{5}$

Step 2

Simplify.

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Step 2.1

Multiply $3$ by $1.1$ .

$3.3 \cdot 10^{2} \cdot 10^{n} = 3.3 \cdot 10^{5}$

Step 2.2

Multiply $10^{2}$ by $10^{n}$ by adding the exponents.

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Step 2.2.1

Move $10^{n}$ .

$3.3 \cdot (10^{n} \cdot 10^{2}) = 3.3 \cdot 10^{5}$

Step 2.2.2

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

$3.3 \cdot 10^{n + 2} = 3.3 \cdot 10^{5}$

Step 3

Divide each term in $3.3 \cdot 10^{n + 2} = 3.3 \cdot 10^{5}$ by $3.3$ and simplify.

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Step 3.1

Divide each term in $3.3 \cdot 10^{n + 2} = 3.3 \cdot 10^{5}$ by $3.3$ .

$\frac{3.3 \cdot 10^{n + 2}}{3.3} = \frac{3.3 \cdot 10^{5}}{3.3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3.3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3.3 \cdot 10^{n + 2}}{3.3} = \frac{3.3 \cdot 10^{5}}{3.3}$

Step 3.2.1.2

Divide $10^{n + 2}$ by $1$ .

$10^{n + 2} = \frac{3.3 \cdot 10^{5}}{3.3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide using scientific notation.

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Step 3.3.1.1

Group coefficients together and exponents together to divide numbers in scientific notation.

$10^{n + 2} = (\frac{3.3}{3.3}) (\frac{10^{5}}{1})$

Step 3.3.1.2

Divide $3.3$ by $3.3$ .

$10^{n + 2} = 1 \frac{10^{5}}{1}$

Step 3.3.1.3

Divide $10^{5}$ by $1$ .

$10^{n + 2} = 1 \cdot 10^{5}$

Step 3.3.2

Multiply $10^{5}$ by $1$ .

$10^{n + 2} = 10^{5}$

Step 3.3.3

Raise $10$ to the power of $5$ .

$10^{n + 2} = 100000$

Step 4

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

$10^{n + 2} = 10^{5}$

Step 5

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

$n + 2 = 5$

Step 6

Move all terms not containing $n$ to the right side of the equation.

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Step 6.1

Subtract $2$ from both sides of the equation.

$n = 5 - 2$

Step 6.2

Subtract $2$ from $5$ .

$n = 3$