====== Exponential Numbers ======

An exponent is a positive or negative number or 0 placed above and to the right of a quantity.

It expresses the power to which the quantity is to be raised or lowered.

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**Example:** $ 2^3 $ 3 is the exponent. It shows 2 is to be used as a factor three times $ 2 \times 2 \times 2 $.

$ 2^3 $ read as two to the third power (or two cubed).

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$ x \neq 0 $ olmak üzere $ x^{-\alpha} = \frac{1}{x^\alpha} $'dır.

If the exponent is negative, the number is written to the denominator and changes sign.

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**Example:** 
$$ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $$
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  * All powers of positive numbers are positive.
  * Even powers of negative numbers are positive and odd powers are negative.
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$$ x \neq 0 \quad \land \quad x \in R \quad \Rightarrow \quad x^{0} = 1 $$

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$$ (-2)^{0} = 1 $$
$$ (-\frac{1}{3})^{0} = 1 $$

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$ 0^0 $ is ambiguous.
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$$ (x^y)^z = x^{y \cdot z} $$

When taking the exponent of an exponent, the exponents are multiplied by each other.
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$$ (\frac{x}{y})^z = \frac{x^z}{y^z} $$

When taking the power of a fraction, the power of the numerator and denominator are taken separately.
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===== Operations with exponents =====

==== Addition and Subtraction ====

When adding or subtracting exponents with the same base and exponent, the coefficients are added or subtracted.

==== Multiplication and Division ====

When multiplying exponents with the same base, the exponents are added.

$$ x^a \cdot x^b = x^{a+b} $$

When multiplying numbers with different bases and the same exponent, the bases are multiplied and the exponent remains the same.

$$ x^a \cdot y^a = (x \cdot y)^a $$

When dividing exponents with the same base, the exponent of the numerator is subtracted from the exponent of the denominator.

$$ \frac{x^a}{x^b} = x^{a-b} $$

When dividing numbers with different bases and the same exponent, the bases are divided and the exponent remains the same.

$$ \frac{x^a}{y^a} = (\frac{x}{y})^a $$

===== Ordering in exponents =====

  - Among exponential numbers with the same base, the one with the larger exponent has the larger value.
  - Among the numbers with different bases and the same exponent, the one with the larger base has the larger value.


===== Examples =====

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==== Example 1 ====

$$ 4^x - 4^y = 768 $$
$$ 2^x - 2^y = 16 $$

What is **$ x+y $ ?**

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<WRAP column>**A)** 6</WRAP>
<WRAP column>**B)** 8</WRAP>
<WRAP column>**C)** 9</WRAP>
<WRAP column>**D)** 10</WRAP>
<WRAP column>**E)** 12</WRAP>
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<hidden Show solution>
\begin{align}
4^x - 4^y = 768 & \\
2^x - 2^y = 16 & \\
(2^x - 2^y) (2^x + 2^y) = 768 & \\
(2^x + 2^y) \cdot 16 = 768 & \\
(2^x + 2^y) = 48 & \\
\end{align}

\begin{align}
2^x + 2^y = 48 & \\
\underline{+\quad 2^x - 2^y = 16} & \\
2 \cdot 2^x = 64 & \\
2^x = 32 & \\
x = 5 & \\
y = 4 & \\
x + y = 9 & \\
\end{align}

The correct answer is **C**.
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==== Example 2 ====

$$  \frac{(0.008)^{0.\overline{3}}}{0.4} = 2x $$

What is **x?**

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<WRAP column>**A)** $ \frac{1}{8} $</WRAP>
<WRAP column>**B)** $ \frac{1}{4} $</WRAP>
<WRAP column>**C)** 1</WRAP>
<WRAP column>**D)** 2</WRAP>
<WRAP column>**E)** 4</WRAP>
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<hidden Show solution>

$$ 0.\overline{3} = \frac{3}{9} = \frac{1}{3} $$
\begin{align}
\frac{\bigl((0.2)^3)^{\frac{1}{3}}}{0.4} = 2x \\
\frac{0.2}{0.4} = 2x \\
\frac{2}{4} = 2x \\
\frac{1}{4} = x \\
\end{align}
The correct answer is **B**.
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