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.

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).

$ 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.

Example: $$ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $$

All powers of positive numbers are positive.
Even powers of negative numbers are positive and odd powers are negative.

$$ x \neq 0 \quad \land \quad x \in R \quad \Rightarrow \quad x^{0} = 1 $$

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

$ 0^0 $ is ambiguous.

$$ (x^y)^z = x^{y \cdot z} $$

When taking the exponent of an exponent, the exponents are multiplied by each other.

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

When taking the power of a fraction, the power of the numerator and denominator are taken separately.

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

Example 1

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

What is $ x+y $ ?

A) 6

B) 8

C) 9

D) 10

E) 12

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.

Example 2

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

What is x?

A) $ \frac{1}{8} $

B) $ \frac{1}{4} $

C) 1

D) 2

E) 4

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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