Exponents and Scientific Notation

1. a) When multiplying powers with the same base, add exponents:

$2^5 \times 2^3 = 2^{5 + 3} = 2^8 = 256$

b) When dividing powers with the same base, subtract exponents:

$7^9 \div 7^4 = 7^{9 - 4} = 7^5 = 16,807$

c) When raising a power to a power, multiply exponents:

$(3^2)^4 = 3^{2 \times 4} = 3^8 = 6,561$

d) Any nonzero number to the zero power equals 1:

$5^0 = 1$

2. a) Move the decimal after the first nonzero digit:

$48,000,000 = 4.8 \times 10^7$

b) Move the decimal 4 places to the right to make 7.2:

$0.00072 = 7.2 \times 10^{-4}$

c) Move the decimal 5 places left:

$305,000 = 3.05 \times 10^5$

3. a) $10^5$ means move the decimal 5 places right:

$6.1 \times 10^5 = 610,000$

b) $10^{-3}$ means move the decimal 3 places left:

$4.08 \times 10^{-3} = 0.00408$

c) $10^0 = 1$, so:

$9 \times 10^0 = 9$

4. Multiply the decimal factors and the powers of 10 separately:

$(2.5 \times 10^{-2}) \times (3 \times 10^4)$

$2.5 \times 3 = 7.5$

$10^{-2} \times 10^4 = 10^2$

So the product is:

$7.5 \times 10^2$

In standard notation:

$7.5 \times 10^2 = 750$

The approximate total mass is $7.5 \times 10^2$ grams, or 750 grams.

5. Compare the powers of 10 first:

$10^7$ is 10 times as large as $10^6$.

Even though 7.2 is greater than 6.9, the second number has the larger power of 10:

$6.9 \times 10^7 > 7.2 \times 10^6$

So $6.9 \times 10^7$ is greater.