QuestionSimplify these expressions using positive exponents: a. $2^{-9} \cdot 2^{-6}$ , b. $2^{9} \cdot 2^{6}$ , c. $5^{15} \div 5^{3}$ , d. $5^{15} \div 5^{-3}$ , e. $(-3)^{-2}$ , f. $\frac{b^{2}}{b^{-3}}$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with exponential notation. . The base numbers are the same in each expression.
3. The rules of exponents apply.

STEP 2

The rule of exponents states that when you multiply two powers with the same base, you add the exponents. So, we add the exponents in the expression $2^{-9} \cdot2^{-6}$ .
$2^{-9} \cdot2^{-6} =2^{-9 + (-6)}$

STEP 3

Now, perform the addition operation in the exponent.
$2^{-9 + (-6)} =2^{-15}$

STEP 4

The rule of exponents states that a negative exponent means that the base is on the wrong side of the fraction line, so you flip the base to the other side. So, we rewrite $2^{-15}$ as a fraction with a positive exponent.
$2^{-15} = \frac{1}{2^{15}}$ This is the simplest form of $2^{-9} \cdot2^{-6}$ with positive exponents.

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