Question

1. multiply. write each product in scientific notation.

(a) $(4 \times 10^{-5})(2.1 \times 10^{-7})$
(b) $(4 \times 10^{5})(6 \times 10^{7})$
for problems 23 and 24, solve and check the equation. show each step.

1. $d - 8.47 = 9.1$
2. $0.25m = 3.6$

simplify:

1. $\frac{3 + 5.2 - 1}{4 - 3 + 2}$
2. $1\\,\text{kg} - 75\\,\text{g}$
3. $3.7 + 2\frac{5}{8} + 15$ (decimal answer)
4. (a) $(-5) - (-2)(-3) - (+4)$

(b) $\frac{(-3) + (-3)(+4)}{(+3) + (-4)}$

1. (a) $(3x)(4y)$

(b) $(6m)(-4m^2n)(-mnp)$

1. collect like terms: $3ab + a - ab - 2ab + a$

Explanation:

22(a)

Step1: Multiply coefficients and exponents

For \((4\times10^{-5})(2.1\times10^{-7})\), multiply the coefficients \(4\) and \(2.1\), and add the exponents of \(10\) (using \(a^m\times a^n=a^{m + n}\)).
\(4\times2.1 = 8.4\), and \(-5+(-7)=-12\).

Step2: Write in scientific notation

So the product is \(8.4\times10^{-12}\).

Step1: Multiply coefficients and exponents

For \((4\times10^{5})(6\times10^{7})\), multiply coefficients \(4\) and \(6\), add exponents of \(10\).
\(4\times6 = 24\), \(5 + 7=12\), so we have \(24\times10^{12}\).

Step2: Adjust to scientific notation

Since \(24 = 2.4\times10^{1}\), then \(24\times10^{12}=2.4\times10^{1}\times10^{12}=2.4\times10^{13}\) (using \(a^m\times a^n=a^{m + n}\)).

Step1: Solve for \(d\)

To solve \(d - 8.47=9.1\), add \(8.47\) to both sides of the equation.
\(d=9.1 + 8.47\)

Step2: Calculate the sum

\(9.1+8.47 = 17.57\)

Step3: Check

Substitute \(d = 17.57\) into the original equation: \(17.57-8.47=9.1\), which is true.

Answer:

\(8.4\times10^{-12}\)

22(b)