simplify:
a. $5^{-2}$ b. $3^{0}$ c. $10^{-4}$
write each number in scientific notation:
d. $0.00000025$ e. $0.000000001$ f. $0.000105$
write each number in standard form:
g. $4.5 \times 10^{-7}$ h. $1 \times 10^{-3}$ i. $1.25 \times 10^{-5}$
j. explain in exercises g–i, how did you shift the decimal point as you changed each number from scientific notation to standard form?
compare:
k. $1 \times 10^{-3} \bigcirc 1 \times 10^{2}$ l. $2.5 \times 10^{-2} \bigcirc 2.5 \times 10^{-3}$
m. use digits to write “three point five times ten to the negative eight.”

Question

simplify:
  a. $5^{-2}$  b. $3^{0}$  c. $10^{-4}$
write each number in scientific notation:
  d. $0.00000025$  e. $0.000000001$  f. $0.000105$
write each number in standard form:
  g. $4.5 \times 10^{-7}$  h. $1 \times 10^{-3}$  i. $1.25 \times 10^{-5}$
  j. explain in exercises g–i, how did you shift the decimal point as you changed each number from scientific notation to standard form?
compare:
  k. $1 \times 10^{-3} \bigcirc 1 \times 10^{2}$  l. $2.5 \times 10^{-2} \bigcirc 2.5 \times 10^{-3}$
  m. use digits to write “three point five times ten to the negative eight.”

Accepted Answer

# Explanation (for part a: $5^{-2}$):
## Step1: Recall negative exponent rule
The rule for negative exponents is $a^{-n}=\frac{1}{a^{n}}$ (where $a
eq0$ and $n$ is a positive integer). So for $5^{-2}$, we apply this rule: $5^{-2}=\frac{1}{5^{2}}$.
## Step2: Calculate $5^{2}$
$5^{2}=5\times5 = 25$. So $\frac{1}{5^{2}}=\frac{1}{25}$.
# Answer (for part a): $\frac{1}{25}$

# Explanation (for part b: $3^{0}$):
## Step1: Recall zero exponent rule
The rule for zero exponents is $a^{0}=1$ (where $a
eq0$). So for $3^{0}$, we apply this rule: $3^{0}=1$.
# Answer (for part b): $1$

# Explanation (for part c: $10^{-4}$):
## Step1: Apply negative exponent rule
Using $a^{-n}=\frac{1}{a^{n}}$, for $10^{-4}$, we get $10^{-4}=\frac{1}{10^{4}}$.
## Step2: Calculate $10^{4}$
$10^{4}=10\times10\times10\times10 = 10000$. So $\frac{1}{10^{4}}=\frac{1}{10000}=0.0001$.
# Answer (for part c): $0.0001$

# Explanation (for part d: $0.00000025$ in scientific notation):
## Step1: Recall scientific notation form
Scientific notation is in the form $a\times10^{n}$, where $1\leq|a|\lt10$ and $n$ is an integer. We need to move the decimal point to get a number between 1 and 10. For $0.00000025$, moving the decimal point 7 places to the right gives $2.5$.
## Step2: Determine the exponent
Since we moved the decimal point 7 places to the right, and the original number is less than 1, the exponent $n$ is $-7$. So $0.00000025 = 2.5\times10^{-7}$.
# Answer (for part d): $2.5\times10^{-7}$

# Explanation (for part e: $0.000000001$ in scientific notation):
## Step1: Move decimal point
Moving the decimal point 9 places to the right in $0.000000001$ gives $1$ (which is between 1 and 10).
## Step2: Determine exponent
Since we moved the decimal point 9 places to the right and the original number is less than 1, the exponent is $-9$. So $0.000000001 = 1\times10^{-9}$.
# Answer (for part e): $1\times10^{-9}$

# Explanation (for part f: $0.000105$ in scientific notation):
## Step1: Move decimal point
Moving the decimal point 4 places to the right in $0.000105$ gives $1.05$ (between 1 and 10).
## Step2: Determine exponent
Since we moved the decimal point 4 places to the right and the original number is less than 1, the exponent is $-4$. So $0.000105 = 1.05\times10^{-4}$.
# Answer (for part f): $1.05\times10^{-4}$

# Explanation (for part g: $4.5\times10^{-7}$ in standard form):
## Step1: Recall scientific to standard form
For a number in scientific notation $a\times10^{-n}$ (where $n$ is positive), we move the decimal point $n$ places to the left. Here, $a = 4.5$ and $n = 7$.
## Step2: Move decimal point
Moving the decimal point 7 places to the left in $4.5$: $4.5$ becomes $0.00000045$ (we add 6 zeros before 4.5 to move the decimal 7 places left: $0.00000045$).
# Answer (for part g): $0.00000045$

# Explanation (for part h: $1\times10^{-3}$ in standard form):
## Step1: Apply decimal shift rule
For $1\times10^{-3}$, $n = 3$, so we move the decimal point 3 places to the left in $1$.
## Step2: Move decimal point
Moving the decimal point 3 places to the left in $1$ gives $0.001$.
# Answer (for part h): $0.001$

# Explanation (for part i: $1.25\times10^{-5}$ in standard form):
## Step1: Apply decimal shift rule
For $1.25\times10^{-5}$, $n = 5$, so we move the decimal point 5 places to the left in $1.25$.
## Step2: Move decimal point
Moving the decimal point 5 places to the left: $1.25$ becomes $0.0000125$ (add 4 zeros before 1.25 to move the decimal 5 places left: $0.0000125$).
# Answer (for part i): $0.0000125$

# Brief Explanations (for part j):
To convert from scientific notation $a\times10^{-n}$ (where $n$ is a positive integer) to standard form, we use the negative exponent's meaning. Since $10^{-n}=\frac{1}{10^{n}}$, multiplying by $10^{-n}$ is the same as dividing by $10^{n}$. Dividing by $10^{n}$ means moving the decimal point $n$ places to the left (because each division by 10 moves the decimal one place left). For example, in $4.5\times10^{-7}$, $10^{-7}$ means divide by $10^{7}$, so we move the decimal in $4.5$ 7 places left.
# Answer (for part j): To convert $a\times10^{-n}$ ( $n>0$) to standard form, move the decimal point in $a$ $n$ places to the left.

# Explanation (for part k: $1\times10^{-3}$ vs $1\times10^{2}$):
## Step1: Analyze powers of 10
Recall that $10^{-3}=\frac{1}{10^{3}} = 0.001$ and $10^{2}=100$. So $1\times10^{-3}=0.001$ and $1\times10^{2}=100$.
## Step2: Compare the two numbers
Since $0.001\lt100$, we have $1\times10^{-3}\lt1\times10^{2}$.
# Answer (for part k): $1\times10^{-3}\lt1\times10^{2}$ (the symbol is $\lt$)

# Explanation (for part l: $2.5\times10^{-2}$ vs $2.5\times10^{-3}$):
## Step1: Analyze the exponents
For numbers with the same coefficient ($2.5$ here), we compare the exponents of 10. Recall that for negative exponents, the larger the absolute value of the exponent, the smaller the number. $|-2| = 2$ and $|-3| = 3$, so $10^{-2}\gt10^{-3}$ (because $\frac{1}{10^{2}}=0.01$ and $\frac{1}{10^{3}}=0.001$, and $0.01\gt0.001$).
## Step2: Compare the two numbers
Since $10^{-2}\gt10^{-3}$, multiplying both by $2.5$ (a positive number) preserves the inequality. So $2.5\times10^{-2}\gt2.5\times10^{-3}$.
# Answer (for part l): $2.5\times10^{-2}\gt2.5\times10^{-3}$ (the symbol is $\gt$)

# Explanation (for part m: "three point five times ten to the negative eight" in digits):
## Step1: Translate the words to scientific notation form
"Three point five" is $3.5$, "times ten to the negative eight" is $10^{-8}$. So combining them, we get $3.5\times10^{-8}$.
# Answer (for part m): $3.5\times10^{-8}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall negative exponent rule

Step2: Calculate \(5^{2}\)

Step1: Recall zero exponent rule

Step1: Apply negative exponent rule

Step2: Calculate \(10^{4}\)

Step1: Recall scientific notation form

Step2: Determine the exponent

Answer: