Question

directions: simplify the expression. write the answer in scientific notation.

1. $(1.5 \times 10^{4}) \times (5 \times 10^{4})$
2. $(2.5 \times 10^{3}) \times (3.1 \times 10^{1})$
3. $(9.4 \times 10^{-1}) \div (4.7 \times 10^{-2})$
4. $(6.1 \times 10^{2}) \times (1.5 \times 10^{-2})$

Explanation:

Problem 1: \((1.5 \times 10^{4}) \times (5 \times 10^{4})\)

Step 1: Group coefficients and powers of 10

Using the commutative and associative properties of multiplication, we can group the coefficients (the numbers in front of the powers of 10) and the powers of 10 together:
\((1.5 \times 5) \times (10^{4} \times 10^{4})\)

Step 2: Multiply the coefficients

Multiply \(1.5\) and \(5\):
\(1.5 \times 5 = 7.5\)

Step 3: Multiply the powers of 10

When multiplying powers of 10 with the same base, we add the exponents (using the rule \(a^m \times a^n = a^{m+n}\)):
\(10^{4} \times 10^{4} = 10^{4+4} = 10^{8}\)

Step 4: Combine the results

Multiply the result from Step 2 and Step 3:
\(7.5 \times 10^{8}\)

Problem 2: \((2.5 \times 10^{3}) \times (3.1 \times 10^{1})\)

Step 1: Group coefficients and powers of 10

\((2.5 \times 3.1) \times (10^{3} \times 10^{1})\)

Step 2: Multiply the coefficients

\(2.5 \times 3.1 = 7.75\)

Step 3: Multiply the powers of 10

\(10^{3} \times 10^{1} = 10^{3+1} = 10^{4}\)

Step 4: Combine the results

\(7.75 \times 10^{4}\)

Problem 3: \((9.4 \times 10^{-1}) \div (4.7 \times 10^{-2})\)

Step 1: Group coefficients and powers of 10

Using the properties of division, we can group the coefficients and the powers of 10:
\(\frac{9.4}{4.7} \times \frac{10^{-1}}{10^{-2}}\)

Step 2: Divide the coefficients

\(\frac{9.4}{4.7} = 2\)

Step 3: Divide the powers of 10

When dividing powers of 10 with the same base, we subtract the exponents (using the rule \(\frac{a^m}{a^n} = a^{m-n}\)):
\(\frac{10^{-1}}{10^{-2}} = 10^{-1 - (-2)} = 10^{-1 + 2} = 10^{1}\)

Step 4: Combine the results

Multiply the result from Step 2 and Step 3:
\(2 \times 10^{1}\)

Problem 4: \((6.1 \times 10^{2}) \times (1.5 \times 10^{-2})\)

Answer:

s:

1. \(\boldsymbol{7.5 \times 10^{8}}\)
2. \(\boldsymbol{7.75 \times 10^{4}}\)
3. \(\boldsymbol{2 \times 10^{1}}\) (or \(20\), but scientific notation is preferred)
4. \(\boldsymbol{9.15 \times 10^{0}}\) (or \(9.15\))