We have six problems to solve: 1. Round the number 689,653 to three significant figures.

1. Problem Description

We have six problems to solve:

1. Round the number 689,653 to three significant figures.

2. Subtract $456_8$ from $722_8$.

3. Simplify the expression $3\sqrt{45} - 3\sqrt{5} + 16\sqrt{20}$.

4. Given that $3\log a + 5\log a - 6\log a = \log 64$, find the value of $a$.

5. Factorize the quadratic expression $5p^2 - 2p - 16$.

6. A variable $P$ varies inversely as the square of $Q$. If $P=5$ when $Q=6$, find $Q$ when $P=1.8$.

2. Solution Steps

1. Rounding 689,653 to three significant figures:

The first three significant figures are 6, 8, and

9. The next digit is 6, which is greater than or equal to 5, so we round the 9 up to

1

0. This means we increase the 8 to 9 and the 6 becomes

6

9

0. Then we replace the remaining digits with zeros. Thus, the result is 690,

0

0

0.

2. Subtracting $456_8$ from $722_8$:

$722_8 - 456_8$

Starting from the rightmost digit:

$2-6$ is not possible in base

8. We borrow 1 from the next digit (2), which becomes

1. Borrowing 1 in base 8 means adding 8 to the current digit. So, we have $2+8-6 = 10-6=4$.

Next, we have $1-5$, which is not possible. We borrow 1 from the next digit (7), which becomes

6. Borrowing 1 in base 8 means adding 8 to the current digit. So we have $1+8-5 = 9-5=4$.

Finally, we have $6-4=2$.

Therefore, $722_8 - 456_8 = 244_8$.

3. Simplifying $3\sqrt{45} - 3\sqrt{5} + 16\sqrt{20}$:

$3\sqrt{45} = 3\sqrt{9 \times 5} = 3(3\sqrt{5}) = 9\sqrt{5}$

$16\sqrt{20} = 16\sqrt{4 \times 5} = 16(2\sqrt{5}) = 32\sqrt{5}$

So, $3\sqrt{45} - 3\sqrt{5} + 16\sqrt{20} = 9\sqrt{5} - 3\sqrt{5} + 32\sqrt{5} = (9-3+32)\sqrt{5} = (6+32)\sqrt{5} = 38\sqrt{5}$.

4. Finding the value of $a$ given $3\log a + 5\log a - 6\log a = \log 64$:

$3\log a + 5\log a - 6\log a = (3+5-6)\log a = 2\log a = \log (a^2)$

$\log (a^2) = \log 64$

Since the logarithms are equal, their arguments must be equal:

$a^2 = 64$

Taking the square root of both sides, $a = \pm 8$. Since the logarithm is only defined for positive arguments, $a=8$.

5. Factorizing $5p^2 - 2p - 16$:

We look for two numbers that multiply to $5 \times -16 = -80$ and add up to $-2$. Those numbers are -10 and

8. $5p^2 - 2p - 16 = 5p^2 - 10p + 8p - 16 = 5p(p-2) + 8(p-2) = (p-2)(5p+8)$

6. Finding $Q$ when $P=1.8$, given that $P$ varies inversely as the square of $Q$, and $P=5$ when $Q=6$:

$P \propto \frac{1}{Q^2}$

$P = \frac{k}{Q^2}$, where $k$ is the constant of proportionality.

Using the given values $P=5$ and $Q=6$, we can find $k$:

$5 = \frac{k}{6^2}$

$k = 5 \times 36 = 180$

So, $P = \frac{180}{Q^2}$.

Now, we want to find $Q$ when $P=1.8$:

$1.8 = \frac{180}{Q^2}$

$Q^2 = \frac{180}{1.8} = \frac{1800}{18} = 100$

$Q = \sqrt{100} = 10$.

3. Final Answer

1. A. 690,000

2. B. 244eight

3. A. 38√5

4. C. 8

5. C. (p-2)(5p+8)

6. B. 10