5. find the value of: $2^{-3}$\
6. combine the terms: $\dfrac{x^7 \times x^2}{x^5}$ (1 mark)\
7. combining the terms: $\dfrac{(b^3)^4}{b^2 \times b^7}$ (2 marks)\
8. combine the terms: $\dfrac{6^2 \times 2^3}{4^2}$\
9. convert the following ordinary number into standard form: 56,000 (1 mark)\
10. convert the following decimal into standard form: 0.0000081\
11. write the following as an ordinary number: $4.2 \times 10^6$ (1 mark)\
12. write the following as an ordinary number: $7.9 \times 10^{-4}$ (1 mark)\
13. which is larger? $6.3 \times 10^3$ or $1.2 \times 10^6$ (1 mark)

Question

Accepted Answer

### Question 5: Find the value of $2^{-3}$
# Explanation:
## Step 1: Recall the negative exponent rule
The rule for negative exponents is $a^{-n}=\frac{1}{a^{n}}$ (where $a
eq0$ and $n$ is a positive integer). For $a = 2$ and $n=3$, we have $2^{-3}=\frac{1}{2^{3}}$.
## Step 2: Calculate $2^{3}$
We know that $2^{3}=2\times2\times2 = 8$.
## Step 3: Substitute back
Substituting $2^{3}=8$ into $\frac{1}{2^{3}}$, we get $\frac{1}{8}=0.125$.
# Answer:
$\frac{1}{8}$ (or $0.125$)

### Question 6: Combine the terms $\frac{x^{7}\times x^{2}}{x^{5}}$
# Explanation:
## Step 1: Use the product rule of exponents in the numerator
The product rule of exponents states that $a^{m}\times a^{n}=a^{m + n}$. For $a=x$, $m = 7$ and $n = 2$, we have $x^{7}\times x^{2}=x^{7 + 2}=x^{9}$. So the expression becomes $\frac{x^{9}}{x^{5}}$.
## Step 2: Use the quotient rule of exponents
The quotient rule of exponents states that $\frac{a^{m}}{a^{n}}=a^{m - n}$ (where $a
eq0$). For $a=x$, $m = 9$ and $n = 5$, we have $\frac{x^{9}}{x^{5}}=x^{9-5}=x^{4}$.
# Answer:
$x^{4}$

### Question 7: Combine the terms $\frac{(b^{3})^{4}}{b^{2}\times b^{7}}$
# Explanation:
## Step 1: Use the power of a power rule in the numerator
The power of a power rule states that $(a^{m})^{n}=a^{m\times n}$. For $a = b$, $m = 3$ and $n = 4$, we have $(b^{3})^{4}=b^{3\times4}=b^{12}$.
## Step 2: Use the product rule of exponents in the denominator
Using $a^{m}\times a^{n}=a^{m + n}$ with $a = b$, $m = 2$ and $n = 7$, we get $b^{2}\times b^{7}=b^{2 + 7}=b^{9}$. So the expression becomes $\frac{b^{12}}{b^{9}}$.
## Step 3: Use the quotient rule of exponents
Using $\frac{a^{m}}{a^{n}}=a^{m - n}$ with $a = b$, $m = 12$ and $n = 9$, we have $\frac{b^{12}}{b^{9}}=b^{12-9}=b^{3}$.
# Answer:
$b^{3}$

### Question 8: Combine the terms $\frac{6^{2}\times2^{3}}{4^{2}}$
# Explanation:
## Step 1: Calculate each power
- Calculate $6^{2}=6\times6 = 36$
- Calculate $2^{3}=2\times2\times2=8$
- Calculate $4^{2}=4\times4 = 16$
## Step 2: Multiply in the numerator
Multiply the results of $6^{2}$ and $2^{3}$: $36\times8 = 288$. So the expression becomes $\frac{288}{16}$.
## Step 3: Divide
Divide $288$ by $16$: $\frac{288}{16}=18$. We can also simplify using exponent rules:
- Note that $4 = 2^{2}$, so $4^{2}=(2^{2})^{2}=2^{4}$
- The numerator $6^{2}\times2^{3}=36\times8 = 288$ (or using exponents, but $6$ and $2$ are not the same base, so direct calculation is easier here). But if we rewrite $6^{2}=36 = 4\times9=2^{2}\times9$, then $6^{2}\times2^{3}=2^{2}\times9\times2^{3}=9\times2^{2 + 3}=9\times2^{5}=9\times32 = 288$, and $4^{2}=2^{4}$, so $\frac{2^{5}\times9}{2^{4}}=2^{5-4}\times9=2\times9 = 18$.
# Answer:
$18$

### Question 9: Convert $56000$ into standard form
# Explanation:
## Step 1: Recall the standard form definition
Standard form (scientific notation) is written as $a\times10^{n}$, where $1\leq|a|\lt10$ and $n$ is an integer.
## Step 2: Determine $a$ and $n$
For $56000$, we move the decimal point 4 places to the left to get $a = 5.6$ (since $56000=5.6\times10000$ and $10000 = 10^{4}$). So $56000 = 5.6\times10^{4}$.
# Answer:
$5.6\times10^{4}$

### Question 10: Convert $0.0000081$ into standard form
# Explanation:
## Step 1: Recall the standard form definition
Standard form is $a\times10^{n}$ with $1\leq|a|\lt10$ and $n$ integer.
## Step 2: Determine $a$ and $n$
For $0.0000081$, we move the decimal point 6 places to the right to get $a = 8.1$. Since we moved the decimal point to the right, $n$ is negative. So $0.0000081=8.1\times10^{-6}$.
# Answer:
$8.1\times10^{-6}$

### Question 11: Write $4.2\times10^{6}$ as an ordinary number
# Explanation:
## Step 1: Recall the meaning of $10^{6}$
$10^{6}=1000000$ (1 million).
## Step 2: Multiply
Multiply $4.2$ by $1000000$: $4.2\times1000000 = 4200000$.
# Answer:
$4200000$

### Question 12: Write $7.9\times10^{-4}$ as an ordinary number
# Explanation:
## Step 1: Recall the meaning of $10^{-4}$
$10^{-4}=\frac{1}{10^{4}}=\frac{1}{10000}=0.0001$.
## Step 2: Multiply
Multiply $7.9$ by $0.0001$: $7.9\times0.0001 = 0.00079$. (We can also move the decimal point 4 places to the left: $7.9\times10^{-4}$ means moving the decimal in $7.9$ four places to the left, so $0.0007.9$ (wait, correct way: $7.9$ with decimal moved left 4 times: $0.00079$).
# Answer:
$0.00079$

### Question 13: Which is larger: $6.3\times10^{3}$ or $1.2\times10^{6}$
# Explanation:
## Step 1: Convert both to ordinary numbers (or compare exponents)
- For $6.3\times10^{3}$: Using $10^{3}=1000$, we get $6.3\times1000 = 6300$.
- For $1.2\times10^{6}$: Using $10^{6}=1000000$, we get $1.2\times1000000 = 1200000$.
## Step 2: Compare the two numbers
Since $1200000>6300$, $1.2\times10^{6}$ is larger. We can also compare the exponents: $10^{6}$ is a larger power of 10 than $10^{3}$ (since $6>3$), and $1.2>0$, so $1.2\times10^{6}$ will be larger than $6.3\times10^{3}$.
# Answer:
$1.2\times10^{6}$ is larger.

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QUESTION IMAGE

Question

Explanation:

Question 5: Find the value of \(2^{-3}\)

Step 1: Recall the negative exponent rule

Step 2: Calculate \(2^{3}\)

Step 3: Substitute back

Step 1: Use the product rule of exponents in the numerator

Step 2: Use the quotient rule of exponents

Step 1: Use the power of a power rule in the numerator

Step 2: Use the product rule of exponents in the denominator

Step 3: Use the quotient rule of exponents

Answer:

Question 6: Combine the terms \(\frac{x^{7}\times x^{2}}{x^{5}}\)